My great uncle Ed was there, at Omaha beach. He survived, but was later wounded in the break out at St. Lo. The sound of the MG42s stuck in his head until the day he died a few years ago.
Anyway, people did strange things back then and some still do today. At the time it was a citizen Army, lots of draftees and recent volunteers. It was a different creature from today's professional all volunteer force. There was a TO&E that regulated who would have what weapon in theory, but it could be a little loose. There were still plenty of officers with their own weapons, substitute standard weapons, etc. Discipline was different since people weren't worried about damaging their career so much as survival. Thus when a "punishment" could entail a nice warm jail cell instead of some time on the front line, the sorts of things that commanders would ignore were different.
Later in the war, my father - who was a combat engineer with the Third Army - said that his people (he was an officer) carried M1 rifles with 48 rounds of ammo in a bandolier. They apparently didn't use the rifleman's belt system, or at least didn't use it to carry ammo. It was a bit uncomfortable and soldiers tended to carry things like fish hooks or cigarettes in the pockets instead of ammunition.
On D-Day itself, one of the problems was that assault troops did in fact go in overloaded and weighed down. They had M1 rifles, four to six grenades, and around 128 rounds of M1 ammo. Handguns and some SMGs along with carbines (and even some 1903 Springfields) were scattered about along with crew served weapons. The men also were loaded down with rations, water, explosives, etc.
As in the rest of the amphibious operations of the war this proved to be a mistake. It caused men to flounder in the surf, be delayed going up the beach, etc. Later in the war "assault" infantry tended to carry as little as possible when actually making an assault. The Marines, for their part, eventually suggested that under fire, a man carry no more than 1/3 body weight in weapons and gear. Given the size of the average man at the time, that would have been around 50 pounds tops, yet on DDay many men were carrying 100 pounds or more of gear. (This problem persists today in someways, perhaps more so since modern body armor is heavy.)
The men of the 82nd and 101st did indeed festoon themselves with weapons - pistols, knives - usually several, rifles, carbines, SMGs, brass knuckles and anything else that was handy.
In other units, it could go either way. Some people took solace in securing extra weapons. Other people discarded as much as possible and carried as little as possible. Even in elite units you'd find both types (google MACV SOG and "mad dog" for a later example, there was a guy that used to carry multiple Smith and Wesson .38s.)
For the purposes of this hypothetical, I'll pick a Colt 38 Super 1911 auto, a UD-42 9mm SMG (made by Marlin) and a Colt 1903 pocket Model 32 and parachute in with the OSS some days before. (And all those guns were in fact purchased by and issued by the OSS...)
Going up the beach? I wouldn't carry a handgun, at least not exposed. At the time it was felt that this marked one as an officer or someone important and drew sniper's fire. I'd tape the spoons down on my grenades (like a smart NCO made all the soldiers do in order to prevent them blowing themselves up accidentally), then I'd obtain an M1 rifle, two 48 rd bandoliers of black tip AP ammo and a utility pocket knife (with can and bottle openers and a corkscrew if possible). I'd throw everthing else away save for some water, a field dressing, zippo, smokes, and some cherry picked goodies from rations. I'd throw out the D ration chocolates since those were tooth breakers, wouldn't carry the period C rations since they were heavy and ungainly, and try to pick only the most edible parts out of a K ration or two and pocket these. I'd also have a 1910 E tool (shovel) or a folding German E tool if I could have gotten one from a vet of N. Africa. Sharpen an edge and it is a good weapon, but more importantly at the time, you could dig holes with it.
Thus M1 rifle, pocket knife, shovel....
