Battlefield captured enemy weapons were shrugged off, but few grunts actually had the time or inclination to loot dead enemy soldiers. US Government or Allied weapons were required to be turned in if recovered on the battefield. Did some folks steal theirs? Sure. But troops were required to stand inspection (it's called a "junk on the bunk", and there would have been few places to stash contraband since footlockers were inspected as well). Many found that when they returned stateside, they could buy those weapons just a few years later for a fraction of what they cost during the war.
I was told by my dad (who served in the Pacific in ww2) that people would pull up alongside the ships in boats and sell any kind of weapon you wanted (swords, rifles, pistols, helmets, etc) for a few packs of smokes or chocolate bars. That way you could show the folks back home what a hero you were. And of course, there was a huge milsurp market in the 50s, 60s and 70s.