That depends on the state, so you will need to check your state's laws. For instance, in my state of Indiana individual face-to-face sales of shotguns and rifles between Indiana residents do not need to go through an FFL, hence no paper trail, although I would strongly advise both buyer and seller to have a bill of sale, but no 4473 must be done. Handguns, on the other hand, MUST be transferred through an FFL and a 4473 filled out. In a neighboring state, KY, it used to be any resident face-to-face sale of any firearm did not need to go through an FFL, but IIRC handguns now need to be transferred, but I am not 100% sure on KY's law on that aspect. Call your local gun shop and ask them, they should know, and then confirm with another gun shop and the police department (but don't take the police’s word as gospel, they are often very wrong about laws regarding firearms that don't involve real victim type crimes). If all three stories are the same, then I would think you would be safe to proceed. If you cannot get straight answers, you may need to contact your state's Attorney General (?) or other official.