Under Federal Law, your FFL (Federal Firearms License) dealer must keep a copy of your purchase paperwork (yellow form) FOREVER. If they go out of business, they must turn the paperwork over to the BATF. This is the same for all states, but individual states have laws regarding local background checks, permits and waiting periods.
Here in Indiana, you can pick up your firearm immediately after the NICS check is approved. A few years ago, a state CCW permit was enough to buy and go without state checks, but they changed that.
As for the background checks, the NCIS system was supposed to destroy approval data after 24 hours. Of course, the RATS decided that they should keep the data for longer in order to perform "audits" of the system's effectiveness. This amounted to a registration, of sorts.
I've always believed that records of guns purchased from a FFL dealer were filed away with my name, probably forever. But I never worried about it, given the quality and dedication of federal agency employees, these copies were probably lost a long time ago.
That's the problem, whether the the government has the records, or the FFL, those records are there forever. In the future, if the anti-gun people get back in charge, they will just order that all FFLs turn in those sheets.
Make no mistake - there are no effective privacy protections for gun owners. This new law doesn't close that gap.
If you purchase two or more firearms from a single dealer within a weeks time, they are required to fill out a form which identifies the purchaser and contains the serial numbers of the firearms and fax it to that BATF.
If that concerns you. Don't buy multiple firearms at the same time from the same dealer.