Eric Schmidt, as Google CEO, had a habit of saying unpopular but true things. When he responded to a question about how people interact with Google with "If you have something that you don't want anyone to know, maybe you shouldn't be doing it in the first place," he wasn't so much offering self-criticism as a warning regarding the overall societal trend, of which Google was merely a part.
The context is often left out in that interview snippet, because he goes on to say "... but if you really need that kind of privacy, the reality is that search engines including Google do retain this information for some time, and it’s important, for example that we are all subject in the United States to the Patriot Act. It is possible that that information could be made available to the authorities."
Actions online leave a data trail. The US government has broad subpoena power to acquire that data trail. DDG is a smaller aggregation house, so it's less likely to be hit first, but I wouldn't assume that they don't retain routing info, IP addresses, etc. that could (with some difficulty) be collected and aggregated to narrow the search space for a person.