Federal Firearms License (FFL) is different than a conceal carry permit (CCP). A CCP will now allow an Alabamian to buy from a FFL dealer with no NICS check. To answer your question, the non-FFL (non-licensed) buyer who possesses a CCP still has to fill out form 4473 and provide government issued ID to prove who they are, again this is in addition to having a CCP. The FFL just doesn't have to run it thru NICS.
I think you are confusing a CCP holder buying from an FFL with a FFL buying from a FFL. Two completely different scenarios.