That depended on many things, mostly logistics. Paratroopers are known to be issued smaller weapons because that's easier to handle when they drop. Thompsons is mostly an officers weapon while BAR is the squad support weapon of it's day. So you had a NCO with a thompson, a support with a BAR and possibly a marksman. The rest would be issued M1 Garands. As for who would be the marksman, that would be down to talent shown at basic training and the like. The BAR and the like are heavy so most likely the stronger recruits would be trained to use them. You know common sense. However there exceptions.

In the pacific I know of at least one instance that the marines demanded bolt action rifles over the M1 Garand and they got them too.

But overal it's logistics and personal talent. Great shooters would be marksman or even snipers, the heavier guys would become the support gunners, support personel like tank-drivers, engineers, mechanics etc would be using lighter weaponry like M1 Carbines that weight less as they're not used as much. Paratroopers would use shorter rifles because that was easier to handle when jumping out of a plane.