I think they really clamped down on personal weapons during the Viet war, for social reasons that I can't address here. Moreover, Hollywood and Congress were gravitating toward gun control as a desired goal.
If you read books about America's wars, you can sometimes see what the authors saw, or carried themselves
Two quick examples are Ted Lawson's, Thirty Seconds Over Tokyo and the several titles from the late Donald R. Burgett.
Lawson said that his B-25 crew had both issued and personal guns and knives. He had a .45, plus his wife's Colt .32 auto. No, he didn't say which make of .45 auto. (I saw the other thread where Ray in Rio speculated that it might have been a Singer.)
Burgett carried a nickel Colt .45 auto, a gift from his father, plus a P-38 captured soon after his paratroop regiment landed in Normandy. Some of them found a supply truck full of P-38's and grabbed them.
During surgery back in England, the wounded Burgett begged doctors not to take his guns, emphasizing that they were his own and meant a great deal to him. A senior doc had them unloaded and hidden under sheets as he was wheeled into surgery. I think he still owned both until his recent death. He also commented on another man or two who carried personal arms. But one lost a nice rifle after being wounded. Burgett said that he carried an issued shoulder arm, as it was more likely to be damaged or lost in battle, esp. if the bearer was wounded and evacuated. He preferred the Garand, as it'd kill with a single torso shot, not true of the .30 carbine in many cases. He found the Thompson too heavy and too short in range. He was issued one as a sergeant/squad leader, but swapped it out to an impressionable private for his Garand.
I think officers could own their pistols until pretty recently.In his series of books about aviator Jake Grafton, former A-6 Intruder pilot and attorney Stephen Coonts armed Grafton with a S&W M-19. He owned an old M-1911 handed down in the family, but didn't carry it in war zones. I think other pilots had personal guns. See the movie of his, Flight of the Intruder, too. I think he carried the same M-19 as Grafton did. Coonts said that pilots had pistol lockers in their cabins. That may have applied to all officers on his carrier.
Gun writer Gene Gangarossa said that he wore a P-38 as a Naval helicopter crewman. I believe that aircrew could carry what they wanted. May not have applied to all carriers.
I read an article by an AF officer who was told to bring his own sidearm to Vietnam, as the unit was short of them.
Famous author Robt. C. Ruark was issued a .45 auto in the Navy and escaped with it and a typewriter, for which the USN eventually billed him. He also acquired a P-38 in Italy, with which he killed a German soldier in an alley.
My mother once had a refrigerator repaired by a former WW II tanker. He told me that everyone in the tank had not only a .45, but some German pistol. He kept his Luger after the war.
Even when regulations forbade personal guns, local commanders sometimes turned a blind eye to them, knowing that they were good for the men's morale.
In Iraq, my son had to carry issued M-4 carbines and a Beretta M-9. Later, going back as a contractor, he could carry what he could find or was in his employer's armory. He chose a Browning Hi-Power MK III 9mm, an M-4, and bought a H-K G-3 7.62mm rifle. which he had to leave behind. He had used a Browning 9mm since his early teens and was very familiar with it. The MK III he had in Iraq had Belgian markings, not US commercial ones. I've seen it in closeup photos. He often wore it on his vest, where it was handy to draw while seated in a vehicle. That way, he could easily shoot a terrorist running up to throw a grenade into the car.
Once, when he wasn't in the vehicle, a jihadi ran up and knifed his Kurdish machine gunner in the leg. The wound was so severe that the man had to leave the job.