I'm not clear whether you're talking about Google and searching using a typical search engine, or were you using ChatGPT on the OpenAI website?

In any case, both utilize publicly available information and databases to form their responses. Searching on a search engine or querying ChatGPT doesn't give them additional information, as that's not how either of these technically work. The querying engine and chatbot are completely separate from the data collection side of the programming.

Having said that, both search engines and AI chatbots use queries to help inform their future data collection of publicly available information. They can use this data as signals, not directly. So if a billion people start asking about "Justin Bieber," both kinds of systems will note that and start looking for more public sources to better help answer people's specific questions about him. (How AI actually works is far, far more complex because it doesn't really understand "information," just word association probabilities, and so this is a greatly simplified version.)

TLDR: Search engines and chatbots don't care what you query and don't use any information you share in a query to then directly insert into their databases.