I don't quite see how AI assisted searching would be different in terms of privacy from normal search engine from the searcher's perspective. Biggest issue with AI in the context of privacy is that it gets trained on a lot of personal data and can also be used to deanonymise when used against others. The only kind of information you would give away to the third party uniquely in AI search engine (I think) would be your speech pattern if you are having a conversation with it instead of simply typing in keywords. Duck.AI claims to ensure your data is not used to train the AI, but there's no way to verify if the AI provider is holding up that promise. Still it's a good choice since you don't have to log in.

Accuracy is not very relevant in privacy, so it would depend almost solely on what you choose as your provider.

If I were to use one, I would choose whichever you can use without logging in, and gives reasonably accurate information.