Are the people who's sites you're using also okay with you passing URLs on their sites to your search provider? Do you ever accidentally paste confidential text in the wrong field?
Because there is no way to tell if the data is being logged (or, importantly, to prove that it isn't) it is prudent to assume it is, with an infinite retention— will your opinions about sharing todays data change if the search provider's behavior changes tomorrow? if some relevant government changes policy tomorrow? Does the fact that you can't ever take it back concern you?
These are some of the things I think of about it— and are why I wouldn't intentionally use such a feature regardless of how positive I feel about my search provider. And because of these thoughts— especially that I can't really make any binding commitment to future policy, or really guarantee that my policy is being followed (e.g. intelligence agency plants on staff) if I were a search provider I would not run such a service, unless I could figure out a way to build it such that I _couldn't_ turn evil, knowingly or unknowingly today or tomorrow.