Presuming one is resident in the US on a visa like H1B or L1, what kind of paperwork do they need to own a firearm? Presume residency in a gun-friendly state like Washington or Alaska if it matters.
Whether federal law allows non-immigrant alien to possess a gun depends on whether they fall into one of a number of exceptions, such as having a valid hunting license or permit:
An alien admitted to the United States under a nonimmigrant visa is prohibited from shipping, transporting, receiving, or possessing a firearm or ammunition unless the alien falls within one of the exceptions provided in 18 U.S.C. 922(y)(2)...
Even if they do fall into one of the federal law exceptions, they also need to check state law, which may have its own restrictions on non-immigrant aliens and permit requirements. A state can be gun friendly but alien unfriendly, so it is a matter of checking the regulations for each state in question.