Why this worry is reasonable
The concern
It feels like a conversation. It is a logged request.
A search box never invited this. You typed three or four words into it, guardedly, because you knew you were typing into a machine. A chat assistant asks for the whole story, and people give it: the health worry you have not raised with a doctor, the argument with your mother that is now in its third week, the paragraph explaining that your daughter has been struggling since the move — with her name in it, and her school, because the tool asked for context and context is what makes the answer good.
What turns that into a real objection rather than a general unease is that the format is doing the persuading. It writes to you in the second person. It asks a follow-up question. It does not react, does not judge, does not tell anyone else, and there is nobody else in the room. Every signal on the surface says private conversation. Underneath, each message is a network request that leaves your house, arrives at a company's servers with an account attached, and lands in a system built to keep records rather than to forget them.
You do not have to believe anyone is acting in bad faith to think that is a poor trade. The gap between what the thing feels like and what the thing is, is the entire objection, and no statement of intent closes it. A company can be run by careful people and still hold years of your family's most candid writing on machines you will never see, under terms it may rewrite, for reasons it will describe later.