From my own perspective (and certainly dont mean this in a way that's dismissive towards you/your comment), given that those companies lie constantly, I'm not sure why I would trust their own PR about their products as a reliable source of information.
This very article. But since you said you don't believe them ("those companies lie constantly") why would referencing them help here? Either you believe them, and it is settled, or you don't and it is worthless.
Hook up wireshark, you can quite clearly see when Google Mini or an Echo are sending an audio stream since it is a substantial packet-load relative to normal "keep-alive" comms. The only exceptions to that, from what I've seen, are automatic software updates (but traffic is moving down, rather than up).
You could also look at the device teardowns, for example this article has a good overview of the underlying workings (inc. Wake Word detection):
https://developer.amazon.com/blogs/alexa/post/2a32d792-d471-...
The only reference that would be credible would be something from the company, so if you don't believe them, then it wouldn't really matter, would it?
A hobby of mine is seeing how far away from "Hey Google" I can stray. The accessibility feature to beep on wake helps with this.
If by 'conspiracy theorist' you mean someone who suspects the government or corporations of foul play, then don't forget about 'government surveilance', one of the biggest conspiracy theories of the last several decades - until it turned out to be true. (did you dismiss them as well, before snowden put hard evidence behind the reasonable speculation?)
Either way, you shouldn't talk down to people who find minimal, available evidence worthy of speculation. It's not conducive to reasonable discussion.