I've been watching band of brothers lately. As paratroopers I would have thought they would all have smaller rifles, (Thompson's and carbines, primarily), but there are several with full sized Garands as well. Did soldiers get to choose their rifle? Also, who got pistols?
Here's a previous thread on a similar topic
As /u/skidleybop puts it:
Weapons were issued on battlefield role and rank. Typically submachine guns (the Thompson, for example) would be issued either to field officers (Captains, Lieutenants) or to specially designated squads.
As far as how weapons were chosen via the platoon, it came down to the above specification and also your proficiency with different weapons during basic training. Better with a rifle than a Thompson? Well then you'd probably be getting that M1 Garand or a similar long rifle. Likewise, as stated beforehand, your role also played a crucial part. If you were part of a tank crew, it wouldn't make sense to carry around a long rifle, as that'd be too hard to get out of the tank. A smaller, more compact weapon like the M1A1 Thompson would prove far more efficient in this role when the weapon is necessary to be used.
I can't find any legitimate sources on the grounds of pistols/sidearms, but I know they weren't generally carried by GIs. It was mostly high-ranking officers who had the privilege of carrying a pistol. Perhaps someone else can shed better light on this section.
See "bayonetstrength.150m.com" for "tables of organization and equipment"; this describes the equipment and manpower allotted for units of platoon strength to battalions in WW2. Depending on type of unit and echelon, equip & manpower varied. These tables and organizations changed throughout the war to some degree. Very educational site. I am not an expert but thought this would get you started.