I believe the instant check info sent will tell type of weapon such as pistol, rifle shotgun or other (receiver). I could be wrong on that. No serial #.

The form you fill out will have the serial number, make & such along with your name and address. If the ATF does a trace they start at the top with the manufacturer or importer and work their way to the FFL. If private sales are banned there will be a paper trail all the way.