Funny

I'm not entirely sure of the situation with firefox. At the very least, if you are searching for a single word then firefox will check for existing TLD domains that use that word. Those lookups can be seen in DNS logs. If the TLD exists then the browser autocompletes the domain name and the browser redirects to the autocompleted url. If the TLD doesn't exist you get redirected to the corresponding default search provider search results, but the DNS logs still have your TLD lookups.

With chrome and chrome-likes, it doesn't really matter because you've already decided to give up your info. For all we know, Google could be logging your search query directly out of the search box before sending the text to your preferred search engine. It's Google - if you can imagine it, they're probably already doing it.