For a private sale, you just have to have no reason to believe the buyer cannot legally purchase a handgun. Some sellers ask for a photocopy of a drivers license and some sort of Bill of Sale. It is common, but not required. A background check is NOT required.
To whom may an unlicensed person transfer firearms under the GCA? A person may sell a firearm to an unlicensed resident of his State, if he does not know or have reasonable cause to believe the person is prohibited from receiving or possessing firearms under Federal law. A person may loan or rent a firearm to a resident of any State for temporary use for lawful sporting purposes, if he does not know or have reasonable cause to believe the person is prohibited from receiving or possessing firearms under Federal law. A person may sell or transfer a firearm to a licensee in any State. However, a firearm other than a curio or relic may not be transferred interstate to a licensed collector.http://www.atf.gov/firearms/faq/faq2.htm#b1Are there certain persons who cannot legally receive or possess firearms and/or ammunition? Yes, a person who –(1) Has been convicted in any court of a crime punishable by imprisonment for a term exceeding 1 year;A person who is under indictment or information for a crime punishable by imprisonment for a term exceeding 1 year cannot lawfully receive a firearm.
(2) Is a fugitive from justice;
(3) Is an unlawful user of or addicted to any controlled substance;
(4) Has been adjudicated as a mental defective or has been committed to a mental institution;
(5) Is an alien illegally or unlawfully in the United States or an alien admitted to the United States under a nonimmigrant visa;
(6) Has been discharged from the Armed Forces under dishonorable conditions;
(7) Having been a citizen of the United States, has renounced his or her citizenship;
(8) Is subject to a court order that restrains the person from harassing, stalking, or threatening an intimate partner or child of such intimate partner; or
(9) Has been convicted of a misdemeanor crime of domestic violence
(10) Cannot lawfully receive, possess, ship, or transport a firearm.
Such person may continue to lawfully possess firearms obtained prior to the indictment or information.