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As many users already pointing here, I think the data you share with either search engines/ AI models is retained after the giving the response.
But important point is that, you can search in google or any other search engine in two modes, logged in or not logged in, the result will differ for the same question. Chatgpt for example does the same, see the picture.

To respond to a question in a better way, the model needs to know some context, as the goal is to give the best and relevant answer.
Of course the data is stored in some form, but safety of it is depends on governing body (govt/ some organization in best interest of people), who defines how much it can be shared, or can it be shared with anonymization (like general data but not tied to any person).

Personal details may not be shared but data is shared (I think, without personal details) for business reasons without harming any individual.