I should believe to be fairly unusual for two very good reasons...
1) The practice of enlisted men carrying sidearms was frowned upon so much that a regulation was enacted making it a military crime to be found with one. A Fall 1862 order from Braxton Bragg (AoT) officially prohibited wearing of pistols by enlisted men, unless specifically authorized to do so, and requiring they turn in their arms to their friendly neighborhood Ordnance Sergeants. Troops doing so would, in turn, be financially compensated for their arms according to a fixed schedule.
2) That there smoke wagon is HEAVY. Soldiers would go through extreme lengths such as trimming an extra inch or two off their blankets to save weight. So....the private purchase pistol, extra powder, bullet mold, powder flask. This is a decent amount of weight and space taken up in a blanket roll.
That being said....I'd would love for it to be the other way so I had an excuse to buy and wear a Remington