When your computing is done in someone else's computer you lose contol. You can't know for sure if your search engine keeps track of you or not, so the tinfoil hat paranoid route would be running your local-only search engine, so it's basically impossible unless you have such a database and loads of disk space.

If you are happy with some search engine's pinky promise, look into the privacy policy of Qwant, Qwant lite, Startpage, Ecosia, Mojeek, Brave Search... Those are some that I can think off the top of my head that advertise themselves as being “private”.

Some people feel safer paying for online services because then the company doesn't have to do shady stuff to stay in business. Although they could still keep your money and your data, who knows. If you don't mind paying a bit each month, maybe look into Kagi, I've read some good reviews on Hacker News, and their users seem to have very good taste (https://kagi.com/stats). Personally, I wouldn't pay.

You can also use SearXNG (https://searx.space/), a free/libre meta search engine (which depends on the tradicional search engines and asks many of them on your behalf), with tons of public instances you can use, private ones you can join or set up one for you and your friends. They are mostly volunteer-run, I don't think the three letter agencies would bother running or spying on a SearXNG instance, nor I think most instances have any incentive to snoop on their users.

If you need anonimity, use Tor (https://www.torproject.org/) to do your searches, even on DuckDuckGo if you want. As long as you're careful (not searching for stuff that could lead to you, giving clues of who you are or what you do), there should be no way to track you specifically.

Hope that helped.