To the OP's issue - definitely an issue that varies by state. As stated above, NH is pretty relaxed with registering boats, all you really need is Bill of Sale, US or otherwise, doesn't really matter. The logic might be, like 92SupraComp said, that they can find a thief easily if he/she registers the boat. Meanwhile they'll take the money happily. And NH does not issue titles for boats, of any age or size, which makes things dicey for lienholders, but anyway that's how it is.

When we sell a boat to someone from NY or MA, though, there's always something complicated with the paperwork thanks to the nuances of the various DMV/RMV whatever of each state.

The best thing to do is to call wherever it is that you are going to register the boat / trailer before you buy the boat to find out EXACTLY what you need. Thanks to the OP for posting the details on his experience, for that exact reason! Save someone else a headache in the future.