Would either of these BJTs require a heat sink?

Heatsinks are generally for power transistors or other power devices. But adding a heat sink Will help to keep the device cool. In your case, Q1 is dissipating only about 67mW which is safely below the max power rating of 625mW and no heatsinl required. The second transistor is dissipating about 200mW. This too is more than half lower than max power rating which is good an wont require heatsink. Note though that these are DC power though and the overall power (DC +AC) must be lower than max power. Take a look at peak power of Q2 when ac signal is applied at input of your circuit. It should be well below Pmax.

To determine if heatsink is require you need to check if Junction temperature of transistor is below a max of 150C . At this temperature the decide will be destroyed. Though a better temperature is 100C. If at your power dissipation the junction temperature reaches 100C then you require a heatsink. The formula for junction temperature is shown below.

Q2 is dissipating 350 milliWatts. Do you feel lucky?

WRONG !! The VDD is only 9 volts, not 12 volts.

Thus the power is somewhat lower. Thanks for catching that error on my part.

If you blow air on the leads, you may be OK.

To remove heat, the 3 leads are your friends. Particularly the collector lead can be the primary path, with the silicon die sitting on a small porch atop the collector lead, inside the plastic. So have 1cm by 1cm piece of metal, close to the collector-plastic interface.

Can much heat exit the base and emitter leads? No. Because of the gold bondwires hidden inside the plastic. Gold is a good heat conductor, but when used in 25 micron diameters (to save on the cost of gold), or about 1 mil (or 0.001 inch diameter), the thermal resistance of just the bondwire is about 10,000 degree Centigrade per watt. However, the transistor die atop collector mounting "porch" will dump heat into the plastic, and the plastic will dump heat into the base and emitter leads; plastic has Rthermal about 200X that of metal, but there are several cubic millimeters of plastic to offer heat transfer paths.