The classic class-B transistor output stage is more like this:
[ The correct terminology is "complementary push-pull output stage", meaning NPN + PNP ]
Note the diodes to bias the bases just into conduction and the small resistors on the output to
limit the current and stabilize the bias - this removes cross-over distortion (which is horrible).
The values chosen here are for a low power headphone circuit, in bigger amps the output
resistors are much smaller in value.
The input is capacitor coupled so no DC offsets are pushed back to the input, the output is capacitor
coupled so the speaker/headphones don't burn up or end-stop.
With no power to the amplifier stage it will be attenuating the signal, not amplifying it. The
base-emitter diodes in the transistors conduct once there's 0.6V or more available, which accounts
for the break-through of the signal without power applied. Generally its not a good idea to run an
amp like this.
If your volume control/switch is correctly configured the volume is fully down when the switch
operates so you'd never hear break-through.