Bipartisan Background Checks Act of 2021 This bill establishes new background check requirements for firearm transfers between private parties (i.e., unlicensed individuals). Specifically, it prohibits a firearm transfer between private parties unless a licensed gun dealer, manufacturer, or importer first takes possession of the firearm to conduct a background check. The prohibition does not apply to certain firearm transfers or exchanges, such as a gift between spouses in good faith.