Embedded Audio ML engineer here (albeit mostly outside of speech). A modern MEMS microphone uses typically 0.8 mA in full performance mode at 1.8V. Doing basic voice activity detection, which is the first step of a continuous listening pipeline, can be done in under 1 mA. Doing basic keyword spotting is likey doable in 10 mA. But this is only done on the part that the voice activity module triggered on. Lets say that is 4 hours per day. Then basic speech recognition, for buying phrases and categorization, would maybe cost 100 mA. But say only 10% of the 4 hours = 0.4 hours have keywords triggered. That would give a total power budget of (1.824)+(104)+(100*0.4) = 123 mAh per day. A typical mobile phone battery is 4000 mAh. People do not expect it to last many days anymore... So I would say that this is a actually in the feasible range. And this is before considering the very latest in low power hardware. Like MEMS mics with 0.3 mA power consumption or lower, MEMS microphones with built-in voice activity detection, or low power neural processing units (NPU) that some microcontrollers now have.

Why would they need to listen continuously?

Most people don’t speak, and aren’t around people talking to them, for more than a few minutes to hours in a day.

You just need to awake every few secs or mins to see if someone is speaking and go back to sleep if they’re not.