How is it that two electric currents can travel in opposite directions on the same wire, at the same time, without interfering with each other?
in the sense that two telegraph signals (in other words, two electric currents) can travel in opposite directions on the same wire, at the same time, without interfering with each other
This is the point. A telegraph wire is not just a wire that transports a direct current. It transports a signal, which in essence is a voltage or current wave. The current oscillates around a value of \$0~V\$ (or \$0~A\$).
If two current waves travel in the opposite direction, the waves have no trouble passing through each other, just as two sound waves are able to travel in opposite directions in the same medium.
(Here, the blue travels to the left, green to the right, and the red wave is the resulting superposition of them. The red wave is the current/voltage distribution that is measured in the wire over time.)
Instead, the value that you would measure at any point corresponds to the linear sum of the amplitudes of both waves - that is what is called the Principle of Superposition. Waves obey this principle if their underlying equations are linear, i.e. the wave equation doesn't contain higher powers (\$x^2\$, \$x^3\$, etc) of the variable that is "waving". That would be the voltage \$U(x, t)\$ or the current \$I(x,t)\$ in the telegraph case.
Specifically, the equations describing waves in a transmission line even have a name: the Telegrapher's Equation. $$\frac{\partial}{\partial x} U(x,t) = -L \frac{\partial}{\partial t} I(x,t) - R I(x,t)$$ $$\frac{\partial}{\partial x} I(x,t) = -C \frac{\partial}{\partial t} U(x,t) - G U(x,t)$$
\$L, C, G \$ and \$R\$ are properties of the transmission line (inductance, capacitance, conductance and resistance) and determine how the shape of the signal is distorted, attenuated and slowed down while it is traveling down the line. The rest describes how an excitation of the voltage and current propagate through this transmission line along the position \$x\$ and through time \$t\$.
As you can see, all of the quantities of interest (voltage \$U\$ and current \$I\$) appear only in the first power. That has the consequence that if you find two solutions to the wave equation, then also their sum will be a solution to the same equation (even when multiplying them by a factor). So if \$U_1(x,t)\$ and \$U_2(x,t)\$ are solutions to the wave equation, then also $$U(x,t) = \alpha \cdot U_1(x,t) + \beta \cdot U_2(x,t)$$ is a solution, where \$\alpha\$ and \$\beta\$ are factors.
Side note about DC:
Having two currents flow in opposite directions would cancel out their contributions and result in no current. Alternatively, you can convince yourself that a direct current (DC) cannot flow in both directions at once, simply by Ohm's Law:
Say you have a wire with a certain resistance \$R\$. According to Ohm's Law, we need a potential difference of \$U = \varphi_2 - \varphi_1\$ between the ends to cause a current \$I = \frac{U}{R}\$ to flow. The current (composed to moving positive charges) moves from the more positive potential to the more negative potential.
We can now flip the potentials: $$U' = \varphi_1 - \varphi_2 = -U.$$ But since the resistance stays the same, this also flips the current: $$I' = \frac{U'}{R} = \frac{-U}{R} = -I.$$
If we make both potentials equal, there will be no difference and the current will be zero.
The only way to have a current come out at both ends is to have a source in the middle, which is not really interesting.
The physics explanation is that waveguides (including the free space) have orthogonal modes for the two propagation directions. This means that the two signals traveling in opposite directions will not interfere. (This is not an approximation, there will be no interference).
The device which separates "transmitted" and "received" signal is a circulator. It exists also in the optical domain and can be used to implement duplex communication over a single optical fiber. In the RF domain it can be used to implement separation of transmit and receive signals over a single antenna (at same time and at same frequency of course). Practically one often uses different frequencies for transmit and receive, mainly due to technical reasons. The circulator has not perfect isolation and the separation does not work so well for very weak received signals. But if one had a perfect circulator device the arrangement would work.
In the old analog telephone system there was only a single wire pair, yet it was possible to speak and hear at the same time.
TL/DR: A very elementary explanation is that one has voltage and current in a wire and that can be used to carry separate information in two directions. Consider the following:
On one side of the wire there is a controllable voltage source, and the to-be-transmitted information is the instantaneous voltage. On the other side of the wire there is a controllable current source (or better "sink"). The to-be-transmitted information here is the instantaneous current. Clearly, station 1 (the one with the voltage source) can read the signal from source 2 by just measuring the current through the wire. Station 2 can also receive the signal from station 1 by measuring the voltage at the terminals of its current source. So this proves you can transmit information in two directions simultaneously over a single wire pair. And if you doubt that it might not be possible to connect a current source/sink to a voltage source. This is perfectly possible, one can think of the current source/sink just as a special form of an adjustable load resistor.
EDIT: There is also a elementary explanation for waves: A free space wave has an oscillating electric and magnetic (E and H) field. They are oriented with 90° angle in space and have a temporal phase shift of 90°. It is +90° for forward and -90° for backward propagating direction (it may be vice versa depending on the choice of the coordinate system or sign of phase). Also the ratio the magnetic and electric field amplitude is fixed to the wave impedance of the medium (which is 377 Ohm for vacuum). If we now have a back and forward propagating wave we will have the superposition of the electric and magnetic fields everywhere in space and time. However a ideal separation of both waves is possible. Simply speaking: The electric fields will add while the magnetic fields will subtract (due to the total phase shift of 180°). Since the amplitudes of E and H field of each component is have a fixed ratio we can substitute the H for E field (or vice versa) and solve for the two E field amplitudes of the forth and back propagating waves. That demonstrates that ideal separation of the two propagation directions is possible.
And the very abstract physics explanation behind this, is - as I wrote earlier - that the modes corresponding to the two propagation directions are always orthogonal and signals do not interfere.
There's your problem: telegraph signals are not electric currents. (We might as well say that telegraph signals are voltage instead.) Which is correct? Neither.
To solve this, give up on electronics and instead fall back to the physics behind it. Actually the telegraph signals (and even all electrical signals everywhere) are actually electrical energy; same thing as light and radio waves. Signals are changes, and a changing current involves voltage, same as changing voltages involving current. Signals are the watts, not just amperes and not just volts.
Signal-energy behaves differently than currents in circuits. While the energy zips across a circuit, instead the amps or charge-flow does not. Charges just rotate through the loop as a whole, or perhaps wiggle slightly back and forth, but current doesn't fly forwards at lightspeed. Something does fly along at lightspeed though. We measure it and discuss it in terms of watts or "wattage." The amps don't fly fast, the amps are different, amps are the slow motions of the "medium;" that charge-sea found inside every wire. Waves versus medium. Somewhat like sound waves versus wind. Electric current is like wind, while signals are like sound waves. (And of course sound waves are back-and-forth wind! The air wiggles, while the waves propagate forwards.)
How can two independent signals pass across an electric circuit? First ask yourself how two independent sound waves are able to pass through the same region of air. And on a pond, throw two pebbles and ask yourself how two bullseye ripple-patterns pass through each other without interacting. Why doesn't one laser beam block another one whenever they cross? It's just something that all waves can do, if the medium is linear. In a linear system, waves can add and then subtract again, so they cross each other without interacting. It works for light inside an optical fiber. It works for sound inside an organ-pipe. It works for coax cable with pulses going in opposite directions, and it works for telegraph signals propagating at lightspeed across a single pair, a single circuit.
The answer to your question involves the waves-chapter of your physics book. The answer to your particular circuits question opens up an entire fascinating field of electronics: cable reflections and standing waves on wires.
On the other hand, two direct currents cannot occupy the same circuit, since they lose their identity, combining to form a sum-current. (Don't forget that every circuit is a one-turn inductor. Similarly, two different voltages cannot occupy the same capacitor! In both cases they combine, and cannot be un-subtracted again.) Two direct currents can occupy a single wire, whenever that wire is a common section of two otherwise separate circuits. But they do this by adding up to form a third current within that common section. (For example, they might subtract to zero current in that section, if they happened to be equal and opposite. One electron can't actually flow in two directions simultaneously.)
Yet at the same time, two completely independent energy-waves (signals) can propagate across one single circuit. HOW? It involves E and M both, and that contains the secret: to understand it we must look at both wires of the long pair, and we must include voltage as well as current. Your question cannot be answered as long as we concentrate on single wires and currents alone, while ignoring the two wires and the voltage across them.
In a single circuit, current is a closed circle, like a flywheel. It doesn't start out in one place and flow to another (instead it just goes clockwise, CW, or perhaps CCW, much like a drive-belt.) A current in a circuit is like a spun flywheel, a closed loop. But something sure does go one-way, right? Whenever a battery lights a bulb, something must go from the battery to the bulb and not return back to the battery. That something is not current. Instead it's EM energy, where energy-flow is measured in terms of watts; of volts times amps. In a flashlight circuit, the wattage is a fast one-way flow from battery to bulb. But the current is a very slow circular flow. Again, the "signal" going from battery to bulb is made of EM energy, not amperes and not electrons.
So here's the start of your answer: in a single circuit, how can we know which direction the electrical energy is flowing? Simple: look at the value of wattage. Specifically: multiply the volts between the wires times the amps through them. If the result is positive, then energy is flowing in one direction, and if it's negative then it's flowing in the other. With a flashlight, hook up your voltmeter and ammeter so they give positive wattage when we multiply them together. Then, when you remove the bulb and install a battery charger instead, the current reverses, so we have energy flowing backwards, into the battery. (This idea is critical with AC, where if V and I waves are in synch, the energy flows continuously forward, but if V and I are at 180deg, the energy instead flows backward.)
So, on a long cable, with an electrical pulse which has positive wattage, the pulse is zooming along left, while if wattage was negative, the pulse is going right. If we suddenly connect and disconnect the flashlight battery, we're launching an energy-wave off along the two wires. It travels at lightspeed and gets absorbed by the flashlight bulb, which lights up. If we leave the battery continuously attached, then still an energy-wave flows to the bulb, even if there's no ripples at all. That's the first concept in basic waves-engineering: the propagation of electrical energy across circuits ...and the idea that "DC" is really just "AC" at very low frequency.
Back to the start again: how can two signal-pulses fly in opposite directions along the same wire-pair? (Note that it must be a wire pair with volts included. Not a single wire.) It can occur if one of the pulses has positive wattage and goes left, while the other pulse has negative wattage and is going right. One pulse might be composed of positive volts and positive amps, while the other pulse is made from negative volts and positive amps. Both pulses are EM waves.
PS
Aha, I see another approach! (Ignore it if you wish, since sheesh this is long.) Suppose we have two separate circuits, two flashlights, but then we meld together one short wire-section from each? The two circuits have one piece of wire in common. Do they interact? Nope, because inside the common wire, the currents just add and subtract again. Each battery lights up its own bulb independently, because each circuit-loop has its own separate battery-voltage and its own separate loop-current. Yet in that common wire, it seems as if two different electric currents are flowing! They're not, not really, because one "circuit current" is the current in one entire loop, including one battery, bulb, and entire closed ring of conductors. In that combined wire, the two currents get added at one end of the wire, then subtracted again at the other. The two energy-waves in each circuit remain independent, even though the currents in their common wire can add and subtract.
This shows us that the answer to your original question cannot involve a single wire. It only can be answered by backing off and taking a wider view; by also including the voltage across two wires.
This also shows how "linear" versus "nonlinear" works. In the common wire, at one end the two currents have combined by adding together. But then they perfectly subtract apart again at the other end. This lets the two loops remain independent. But what if this didn't happen, and instead the currents in the single wire weren't a simple sum combination? Aha, that would be "NONLINEAR." In that case we couldn't cleanly separate them once combined. The "adding together" at one end of the wire wouldn't be perfectly equal to the "subtracting apart" at the other end, and in that case the two separate circuits would start interacting. One battery would begin lighting the other bulb slightly. The signals of the two circuits would truly mix together.
PPPPS
This sort of question has a long history, and a popular book about it is THE MAXWELLIANS, by BJ Hunt. The infamous Oliver Heaviside figured out that telegraph signals were actually EM waves, but then he was nearly suppressed by William Preece, head of the UK govt telegraphy office, who "knew" that dots and dashes were simply currents, period, end of story and don't ask questions or WH Preece will make you sorry! :) Heaviside used his new EM theory of cable-waves to solve a huge telegraphy problem: for any signals traveling along 100KM telegraph lines, the dots would either disappear or "ripple," and for telephone lines, long distance transmission was completely distorted and impossible. (The problem was found to be wave-dispersion or "chirp," where the low freqs travel faster than the high.) Heaviside's "telegrapher's equation" and his "loading coils" fixed this, letting telegraphy become broadband, even when over immense distances. He single-handedly created long-distance telephone. But Preece quickly halted this heresy by using his political power to start an anti-Heaviside bad-mouthing campaign in the press, and a whispering campaign among engineers. Then in the USA, Pupin of Columbia pretended to invent Heaviside's loading coils, patented them, and made millions via Bell Telephone, while Heaviside remained nearly penniless, not gaining fame until after he'd died. (Heh, a Tesla/Marconi story long before Tesla and Marconi. Pupin even played a big role in Tesla's downfall!) So now you see why I'm in love with the story of telegraph-is-EM-waves. Obsessed. Don't even get me started! Oops, too late. :)