What do you gain through a separate search field, anyway? Like, if you started with a unified search/address bar, what would be the arguments for separating the search bar out into a separate bar?
The reason it's bad to do it at all is because you're destroying user faith.
I get that companies need insight about how users use their products. This is not an instance of that, because companies definitely don't need insight about which websites are visited.
My biggest issue is ending up with search results when I forget to use http:// for a tld that is only accessible from the network I am using.
Other than that, I don't like the idea of automatically sending my browser history to my search provider. To that end, I also immediately turn off phishing protection on all new installs.
You should really look into how phishing protection works in Firefox [1], it's not implemented in a naive way (i.e. does not send the url to a third party), and does not compromise your privacy.
With the combined search/URL bar in Chrome, every single keystroke of every single URL you type into your browser is sent to Google. With a separate search field, you get the advantages of auto-completion of search entries without the privacy issues of leaking every URL you type.
I guess I don't find much of a problem with this--or rather, I don't think "my URLs aren't being sent to Google" is much of an aid in protecting my privacy; any number of intermediary routers could log the same information for all non-HTTPS traffic (which is most websites where the URLs are at-all telling), and personally-interested attackers are just as willing to tap my line as they are to ask Google for my browsing history.
What I'm really saying is, if you think something is important enough to use any sort of a private-browsing mode? Spend the extra few seconds and open up Tor instead.
That's a really black-and-white way of thinking about privacy. If using Firefox reduces the amount of logs Google collects about you by, say, 50%, then it's a pretty major gain, even if you're still leaking half of your web history.
How so? I think of it more like a power-law distribution: there are only a few websites anyone would care that I visit, and they're pretty obviously-so when I think about them. Those, the "signal" among the noise, I use Tor for.
Everything else is the same boring noise everyone else generates. The more noise you put out, the less suspicious you look. Nothing says "this person leads a secret life" to the NSA better than having a blank "file" in their automated-dredging system. You want such a file to be full of mundane detail, such that your data doesn't have a different "shape" under traffic analysis than the average 16-year-old who makes public posts about their plans to hang out with their friends at the mall on Saturday.
I like to maintain a history of things I've searched for. I'll often have to monkey with the terms and the sequence to get anything good out of google for the stuff I look for. Sometimes it's just useful to have it separate.
I rely a lot of searching the history and bookmarks from the address bar while the search box heads directly to the search engine and suggests instant results. I also like the privacy I get for not feeding everything I type into Google.
I find that for this kind of query in Chrome, I end up using keyword search: I focus the address bar, type "goog" and press tab, and then type my search query.
It never occurs to me to do it until I see the error page, though; it's always a bit of a hassle.