They're probably not collecting IP addresses or user cookies, but they will undoubtedly have a log of what their users search for, what results were turned, and

They're probably not collecting IP addresses or user cookies, but they will undoubtedly have a log of what their users search for, what results were turned, and possibly what clickthroughs happened from their search page. You can do all of that without retaining identifying information for the person who clicked through. Worst case, if you go back and forth between google search and DDG, a comparative analysis might be able to identify you or people like you from those logs, but it would require some work.