The form you fill out when you buy a gun stays at the FFL holders place of business, ATF can go to the FFL and inspect their paperwork and such, but the paperwork stays with the gun dealer.
I remember quite a few years ago, the ATF was sued because they were showing up at FFL holders with copy machines and copying these forms, the lawsuit charged that they were building an illegal gun registry. I haven't heard of them attempting this in a long time.
I think the background check consists only of your pertinent information used to identify you and if it's a long gun or hand gun, (no model and SN), the NICS response is proceed or deny.
NRA Lifetime Member Since 1999
"I ask, sir, what is the militia? It is the whole people except for a few public officials." George Mason