Distinguishing between solid spheres and hollow spheres (equal mass)

Let both roll down an inclined plane. The hollow sphere will accelerate slower than the solid one (due to their different moments of inertia).

For a solid sphere, the moment of inertia is $$I = \frac{2}{5}mr^2$$ with mass $m$ and radius $r$.

For a hollow sphere it is $$I = \frac{2}{3}mr^2$$ The hollow sphere therefore has a greater moment of inertia and will accelerate slower under the same torque:

$$M = I \frac{d\omega}{dt}$$ with angular velocity $\omega$ and torque $M$.

If you are not allowed "any equipment" I suppose that eliminates using a ramp and letting them roll down it (@andynitrox's otherwise good answer).

However, assuming your hands are not "equipment", if you took one ball in each hand and tried rotating your hands back and forth as fast as you could, you would find that the frequency you could achieve with the solid ball would be greater - again because of the smaller moment of inertia.

What you did there was to create a torsion pendulum without using torsion wire (namely, using just your hands). Assuming that your hands have roughly the same strength, you can rotate the object with lower inertia faster. If you are in doubt whether your two hands have the same strength, you can swap hands and repeat.

Another interesting variation on @andynitrox's answer is to see whether the balls will start to slide when they go down the ramp. It turns out that the friction force needed to stop a ball on a ramp from sliding is a function of the moment of inertia. You can see a detailed derivation of (most of) this in this earlier answer I wrote. The implication is that the solid ball will roll without slipping on a steeper slope than the corresponding hollow ball - the critical angle is $\tan^{-1}\left(\frac{7\mu}{2}\right)$ for the solid sphere, and $\tan^{-1}\left(\frac{5\mu}{2}\right)$ for the hollow one.

The advantage of the "skid" method is that it would allow you to do the comparison one at a time. If you have the appropriately angled ramp (with angle between the two critical angles) you can tell whether a single ball is hollow or solid without needing to compare speeds. But it would require "equipment" of sorts.

Andynitrox gives a great answer, but it sounds like you are considering a ramp to be equipment. However, because the moment of inertia describes the rotational motion of each object, you actually don't need the spheres to move translationally at all.

If you just take each sphere and place it on a flat surface and try to spin it about the vertical axis (like you would spin a basketball) you can compare the angular velocity, $\omega$ with which each of the spheres rotate. No need to measure quantitatively either; you just need to know that the hollow sphere will rotate more slowly. This is because the hollow sphere has the same amount of mass as the solid sphere, but because the mass is located further, on average,from the axis of rotation, it will not increase its angular velocity as much, for the same amount of torque (your spin).

Even though there will be some uncertainty in the exact amount of torque you provide each sphere, you can run multiple trials. Additionally, the difference in moment of inertia should be enough that you will still notice an unambiguous difference.