It has been suggested that background checks be required for all sales, and the records be stored at licensed dealers. There was a diary about that on dKos recently.
The objection raised was that for some tiny fraction of potential sellers, the visit to the licensed dealer would be onerous. The suggestion to that looks obvious to me. Give the buyer and the seller a choice.
1) The private sale can be through a registered dealer. The dealer will keep a record, and the ATF will not.
2) The private seller may request (and pay for) the background check himself. The ATF will keep the record, including the serial number of the gun.
If the buyer is anxious that the ATF not have a record, the transaction can be through a registered dealer. If the trip to the registered dealer would cause problems, then the background check can be direct. The buyer and the seller decide which issue is more important for them, instead of raising that issue in hypothetical terms.
It might be pointed out that if the NRA claim that only a few sales currently escape background checks is right, then this will affect only a few persons. If the bogeyman of gun confiscation legislation were to be passed, then the ATF will have records of only a tiny minority of guns, and those in the least accessible regions.