I like the DuckDuckGo search engine, but I don't care much for their browsers (mobile nor desktop). I do keep the browser app on my phone so I can generate alias Duck addresses, I do find that feature pretty handy.

As for how private DDG search is, I feel like the best practice for using any search engine privately is to clear browser data when you're done searching with it. That's a hassle though, so it's smart to have a browser dedicated to temporary browsing sessions that you don't plan to go back to later. On Android, browsers like Firefox (and forks like Mull) as well as Cromite allow you to set it so browsing data clears when you exit the browser.