It's not powered up, though. That's why it lasts so long on one battery. The button wakes it up.
Without analysing the firmware you wouldn't be able to confirm for sure what it's doing.
I suppose one exercise for those inclined would be to reprogram it to listen (perhaps only during the daytime) and see how long it lasts. You wouldn't need high fidelity so a low sample rate would be enough.
It's on your own network, so it's trivial to know when it's communicating and with whom. In principle, it could periodically wake and poll a C&C server to see whether it should start streaming audio or not.