Any private individual who wants to sell a gun must follow a variety of Federal and State laws. First, the buyer must be an Illinois resident. Second, buyers must have a FOID. Third, buyers can't have any of the characteristics listed above that a FOID is supposed to screen out, nor can they be: mentally defective, dishonorably discharged, or convicted of any kind of criminal domestic violence.
Fourth, for 10 years after the sale the seller must keep a written record of: the date of the gun's transfer, a description of the gun (make, model and ammunition size), the gun's serial number (or "other identifying information" if there's no serial number), and the buyer's FOID number (a photocopy of the whole card would probably be better).
Fifth, and finally, the seller must wait 24 hours before delivering a long gun to a buyer, and 72 hours for handguns. Private individuals don't have to do background checks like licensed dealers do, but they're still subject to the Illinois waiting periods.