Re: Has anyone actually proved, conclusively
Your personal assistant is listening all the time for your wake word. As soon as it thinks you're talking to it, it's recording, and Amazon and (I believe) Google keep transcripts and audio snippets of these to further train their NN models. See Register passim
https://www.theregister.co.uk/2018/11/14/amazon_echo_recordings_murder_trial/
https://www.theregister.co.uk/2018/05/24/alexa_recording_couple/
https://www.theregister.co.uk/2017/02/23/amazon_police_case/
etc
C.