Private sales should not even exist. Since you are selling a lethal weapon not a paperclip and in some states with no background check required and even no taxation for the sale. Let only gun shops deal them, regulate it proper with registrated serial no. for guns and ammo count and you have a better track on what's out there and who might have it.
Still amazed gun shops didn't militate for it to be stoped, but I guess they did the math that the vast majority of gun owners upgrade and swap their weapons quite a lot so by private sale of the old gun they might buy the new one. But that can be easy done via the shop with a buy-back program.