Yep, but how do you know the person you are selling is using a legit ID? What check are you able to perform during a private sale the confirms the seller is who he/she claims to be? That is the point of having actual background checks for all transactions. If it was required and sellers were honest, criminals would have a much harder time buying guns from legal owners. Otherwise, you have no idea who you are selling to in a private sale.
Not claiming it could or would stop all purchases, but it would be a good start. As it is now there is nothing that even slows the flow of guns to criminals other than it isn't easy to buy a brand new gun from a legit dealer.