DuckDuckGo has been my automatic search engine for a while, on both mobile and desktop. I think it still needs some work though. For some things, like images, Google works far better still. Plus, ranking will differ, and pages I can find with Google (or more likely StartPage) I just can't fin with DDG. So, I'm slowly switching over to StartPage, but it doesn't have search bar integration. Once these sorts of iisues get fixed, I'd hope private search engines will gain more traction.