Most Cowboy Action ammunition is fairly low powered to avoid accidents and damage to the targets.
The Single Action Shooter's Society rules require that pistol ammunition have a muzzle velocity of less than 1,000 fps and must use lead bullets only.
Rifle ammo must have a MV of less than 1,400 fps and have lead bullets only.
Here are the SASS Rules: http://www.sassnet.com/pages/SHBv102.pdf
As to whether modern guns made for cowboy shooting are safe with modern loads.......I would guess that they would, unless they state otherwise. Certainly the ones made by reputable manufacturers like Colt and Ruger would be safe.
You gonna be a cowboy, Merc?