Both have option to turn off history saving. In this case they keep dialog for a short time only.

I’m sure this will prevent Google/Bing from storing everything you type in there on their end for as long as they please.

I see this a bit different.

given how the very purpose of LLMs is to just create sequences of words that are statistically likely to follow each other in a sentence, and how there are countless examples out there of them hallucinating answers including non-existent court cases, or providing authors of articles with a list of articles they have not written and so on, I struggle to see it as anything else.