Some stores will do a full BG check for the reason you stated, because the permits can be faked. Its up to the discretion of the FFL.

If you went to a gunshow, and paid someone money and they sold you a gun, and you never had to show anything, they are NOT an FFL dealer. If they are, they just committed a felony.

I think the problem is people confuse the gun show as like a place were everyone is a merchant licensed and such. Its not true. LICENSED FFL Dealers are required by federal law to do a 4473/BG check on anyone buying a gun used or new.

Person to Person sale requires no background check, no form, no ID , etc.