Pretty much as the title. You’re out in the Old West in the days of Wyatt Earp and Jesse James, on the side of law and order. Obviously you’re going to need some iron on your hip - which peacemaker available at the time would best suit your purposes?
Winchester 1873 rifle. Prefer to do my shootin’ from a distance.
As I understand it most shootings were people getting shot from behind or when they were not prepared like how Wild Bill Hickock was shot while playing poker. So being wary of blind sports and having someone watch your back was more important than the gun your carrying.
That being said I think they carried just about everything, rifles that could quick fire or were on a swivel, sawed off shotguns, revolvers, brass knuckles, and clubs.
So lets just say Black Bart and his gang were in a saloon and he wanted to arrest them. A lawman would have his men surround the place and they would all walk in from different sides with guns drawn.
On my hip, a Colt M1861 Navypistol. In the long gun holster on my saddle, a Winchester Model 1873 lever action rifle. Behind my horse, a Gatling gun.
I like the way you think except I would probably pick the S&W Model 2 or maybe 3 for easier/faster reloading.
If we’re talking early-on, when blackpowder revolvers were it, then the Remington 1858; no fiddly wedge to lose or to get loose, and the (theoretical) potential to reload faster by swapping in a fully loaded cylinder.
Later, when metallic cartridges became common, I’d want a Colt Single Action Army.
However, there’s a lot of merit to getting a pistol chambered in .44-40 (don’t know what kind, as I could get a Remington 1873 lever-action rifle and use the same ammunition in both, which might be really handy for a more roving sort of lawman, like a Texas Ranger.
A Mr. Kobayashi hypothetical which doesn’t allow for WWII weapons?
Bass Reeves used a Winchester .44-40 and Colt single-action .32-30 or .45 so I’d probably imitate him. I’d also make sure I kept a shotgun close.
Yeah, absolutely no question but to have a Winchester 1873.
Might be a couple of different pistols to choose from, but that is THE rifle to have.
(proud owner of a modern Model 1894 Cowboy II in .357 magnum)
I have no opinion, but I texted my son, who I knew would. 
His response:
A short shotgun would be best for those situations where you don’t have time for careful aiming and basically just need to sweep the dirt out the door.
Bass Reeveswas one badass lawman. He’s one of my favorite wild west characters, and he’s little known outside of black history.
Being, likely as not, a riverboat gambler, I’m for the little over/under derringer thingie. Love that little puppy, just as cute as a bug’s ear.
On the other hand, I might be a bartender in a gold camp, and so it’s a sawed-off shotgun behind the bar.
Bartender: “This here’s my sawed-off shotgun.”
Customer: “That is a full-length shotgun if ever I have seen one.”
Bartender: “Well, when I points it at someone and shout, ‘Sod Off!’ generally they does.”
It is extremely rare but it does happen. 
So I’m not allowed a phase plasma rifle (40W range)?
Hey,just what you see,pal!
(/hijack) Where are the 2nd amendment enthusiasts to correct the OP’s use of the inflammatory use of “piece” instead of “firearm”? They will jump all over a gun discussion thread… (/end hijack)
For logistical ease, a carbine and revolver that use the same ammunition.
Thinking it over some more, add a Greener to my list. The one I have right now is one of the Old Man’s capguns but from what I’ve heard the boys cartridge arms are pretty damn good in a close fight.