On the Pietta, they don't always have locking pins in the rear of the frame, especially so the brass framed guns, as they can't drill a hole half in brass and half in steel, the drill bends and skids off the steel.
If there is a lock pin in the rear of the arbor, try driving it in harder with a pin punch and that may tighten the arbor.
If there is no lock, then unscrew the arbor, degrease it and the thread, and (red) Locktite it in place, install the barrel and wedge to keep it straight while the Locktite dries.
If driving the lockpin won't tighten the arbor, you'll have to drill it out, a drill bit about .100 diameter. The pin is the same pin part used in the front of the frame, as a pair, for alignment, which will tell you the diameter drill you will need.
With the pin drilled out, you can unscrew the arbor to clean it and reset it with locktite, then tap the lock pin's hole for a #6-32 or #6-40 set screw to relock it.