Under the regs any equipment captured by US service personnel is the property of the US government as the serviceman was acting as their agent. However, with permission, military personnel were allowed to bring back war trophies up through the Vietnam War. This was typically guns, bayonets, helmets, etc. You just filled out a "capture papers" form and got them approved through your chain of command and you could take them home. Probably many more were just tossed in a box and shipped home, if it got confiscated there were plenty more Lugers where that one came from!
NFA stuff included, for a long time ATF accepted capture papers as evidence they were properly registered, maybe they still do? The weird thing is I have seen a 1945 dated set of regs for war trophies and it specifically prohibits machine guns, live ammo, and explosives from being brought back. OTOH, is some officer rubber stamping a stack of forms gonna know an MP44 is prohibited if he even bothered looking? Or they were allowed previously until someone changed the rules later on? Other conflicts, I don't know how they treated NFA stuff. Sometime after 'Nam the taking of firearm war trophies by individuals was banned, and each conflict had their own rules. Our first trip to the sandbox I knew an EOD guy who brought back some deactivated plastic mines, just a round container really but it was allowed. Last time around another guy I knew said all they were allowed was a single bayonet and they almost strip searched you to make sure nothing else got through, even made him pour out a pill bottle of genuine middle east sand he had scooped up.