"Not only would a service like that be welcomed by private buyers, it would be useful to those who have had guns stolen. Wouldn't you like to know that your stolen guns have been properly entered into the database?"
I'm sure it would be welcomed and useful, to a point. But here's what would happen. Joe buys a guy from somebody he doesn't really know. Gets it home, decides to call in the serial number. It comes back hot. Now what? Do you think average Joe is going to tell the police "Sure come by and pick it up." The police won't give Joe back his $500 he just paid for the gun. The gun is confiscated. So Joe will be out the money, out the gun, and will have to tell what and who he bought it from. That's the problem with doing telephone checks. People, as honest as I'm sure everyone on this list is, just don't do that. BTDT. They'll lose the gun, or it really wasn't my gun I was running for some guy, or they try to take the matter in their own hands. So the idealistic view of running a gun over the phone, sight unseen, will more likely than not end up not getting the stolen gun back to it's rightful owner but most likely will end up getting stolen guns pitched in some lake.
Think about it. It's no different than if Joe would call the police to see if there was a warrant out for Joe. Joe confirms there's a warrant and Joe knows to head out of town. NCIC is full of people who know there's a warrant for their arrest but they aren't sticking around as good citizens to get right with the world. They're gone. Same would happen with stolen guns checked over the phone.
So call the police. They'll send someone out to eyeball the gun. If it's hot then it will be confiscated and returned to the owner. If it's not in the computer, the police go on about their business and you head to the range with your new purchase.