This page seems to contain useful information relevant to New York, which appears to handle titles differently than most other states.
In most states, the lienholder holds the vehicle's title until you've paid it off, and then they send you the title. In NY it looks like the purchaser gets the title immediately, and the lienholder is marked on it. When the loan is paid off, the owner gets a certificate of release of the lien, which he should be able to provide along with the title to anybody he's selling the vehicle to.
I only skimmed the faq in that page, so it wasn't clear to me how one sells a vehicle one hasn't finished paying for yet. It's been many years since I did that in any state, so I don't know. I do know if you do it wrong, then the seller can actually walk away with your money and leave you owing for his loan. Such a sale can definitely be done, but don't automatically just trust that the seller knows how to do it.