I doubt their "signal processing" involves anything more than a microphone and a peak detector.
A more practical side channel attack would probably be through radiated emissions over the mains power. However if you're willing to go that far, a laser microphone is much more effective and simpler.
Depending on how wide is the peak detection window, they might send enough data to detect how long consecutive words are. I have a vague impression that I've read once that this, together with a language model, can be used to guess what's being spoken ~10% of the time or so.
A more practical side channel attack would probably be through radiated emissions over the mains power. However if you're willing to go that far, a laser microphone is much more effective and simpler.