No, subsidized phones aren't the carrier's property while you're in contract. You're liable for an early termination fee if you break the contract (barring the case of "adverse changes" or the like), but you own the phone — or at least the atoms in it.

You still have a reduced "expectation of privacy" in your mobile communications, however, in that you're using your carrier as an intermediary to those communications. Consider pen registers, for example.