Actually, the envisioned change to the private-sellers law is that the prospective buyer and seller would go to a licensed gun dealer, who would hold the gun (for a default period of 10 days, iirc) and initiate the background check of the buyer. If the check came with no issues, the sale would go through; if the check came back with one or more issues, presumably the dealer would tell the buyer (but not the seller) what the problem was, and the buyer could appeal the decision in some meaningful way, if that's what he wanted.
The problem with a purely binary result is that it doesn't give the purchaser much recourse if the check comes back "No." I mean, what is the purchaser supposed to do in that case?