Hi,
Be sure that your bag (with the firearm) is property tagged at check-in. Most all carriers have either a special tag for firearms or one that denotes baggage that require special handling, it will be put on your bag at check-in. Some airlines have a form that you must sign that declares you know and are complying with the laws regarding transporting by air.
(I am assuming the passenger knows and complies with all packing requirements)
Upon arrival in the US (the CBP will be pre-alerted by the carrier thru your check-in disclosre, to the presence of a checked firearm) the bag will be retained either by the carrier or the CBP, and delivered to CBP in a special or secured area apart from you or where regular baggage is delivered.
This is because they (customs) doesn't want you - the passenger - to have access to the firearm while inside a secured area.
When you arrive and after you clear immigration, if you weren't contacted by either airline staff, CBP or local law enforcement, by that point, contact a rep of your airline or a CBP officer in the reclaim hall. Tell them your bag has a declared firearm, and produce the bag tag.
They will show you to the location where the bag will be delivered and the clearance process completed will be finished there.
Listen carefully as they usually have explicit instructions about what to do or will happen *after* you clear customs - but are still on/in the airport itself.
Each airport does this different depending on how it's laid out. Usually you are free to go with your bag (and enclosed firearm) and exit the FIS, but some airports require a CBP officer or other officer to escort you out until you're landside.
One point, if the passenger is ever unclear about what to do, where to go, etc.. Be sure to *ask*. The carriage of firearms in most cases, is very legal... But any infractions of these laws tends to be met with serious consequences. So be sure the passenger knows what to do.
Travel Safe,