As stated, in NH - paperwork, nope. But some people do - had a guy once who put my name on a bill of sale I had to sign, it also said I acknowledged I was not prohibited from owning a gun.
He had been tracked down once with a handgun used in a robbery that he had
originally bought - he wasn't charged, didn't even sell it to the person who actually committed the crime, but understandably takes a cya approach.
To me that's fair - a little documentation held privately among private citizens. I might ask for the same. Some people want a double verification you are a resident, ie to see drivers license and a bill to a nh address. As private citizens we can make our own rules. I don't sell my guns often but I require seeing something similar you already own, basically because for me it's piece of mind that this isn't your first gun bought hastily to do something bad with - I figure if a guy already has a shotgun and he was fixing to shoot his wife he would have already done so.