depends on the situation, If the FFL in the buyers state will accept a "common carrier" shipment from a non FFL, then there is only one fee and you can stipulate in the sale the buyer pays it, but the shipping charges are going to be steep, probably more than a FFL in NH would charge you to handle the transaction.
You ask why?
Because a FFL can ship a handgun via U S Mail, where you and I can only send long arms via the U S Postal Service. If the NH FFL ships it Priority or Express mail, and hits you for a reasonable fee, chances are you are still ahead of the game money wise when you count U.P.S. overnight shipping, which is the only way they will ship it, AND the time it would take you to bring it to a shipping depot. You can't ship a handgun UPS from a retail location.
So If you ship it UPS to a FFL, you are going to get hit maybe 60 bucks and your time, where a FFL on your end might do it for 30 bucks and shipping.
The guy buying the gun pays his FFL their fee for their work, which includes the background check, form 4473, and in MA a FA-10 IIRC.
you can always stipulate the buyer pays ALL FFL and shipping costs, including your FFL fee.