To duplicate factory regulation, Doubles should be shot with bullets same as, or with same profile, and especially same weight as was used in the original factory regulation of that given double, and shot at same, or reasonably close to same velocity as shot when rifle was originally regulated. Using the .375 mag. as an example, some doubles were regulated to shoot, to regulation, bullets of 270 grain, and other rifles were regulated to shoot to regulation bullets of 300 grain (both weights were loaded by cartridge makers, for this caliber). Some doubles will shoot both weights equally well, but not all doubles will do this, in this caliber. If you wish to shoot bullets with different profile than rifle was originally regulated for, or possibly somewhat different weight, you must try them to see what performance you get---sometimes they will shoot to regulation well; sometimes they will not, depending on the given rifle, the different bullet profile, the different weight.
Concerning shooting mono-metal bullets, there were barrels separated from the ribs and also rifling "tracks" showing up on the outside of the barrels, but I think that was rare, and seen only on old vintage softer steel barrels. With modern alloy steel barrels, and with more refined modern mono-metal and steel jacketed bullets, it isn't a problem any more. However, they certainly have to be harder on any and all barrel steel rifling, including the most modern steel ones, than softer lead bullets with copper alloy jackets on same barrels would be. Shooting a few such mono bullets should not be much of a problem, but shooting a lot of them is bound to wear out a barrel faster than if only lead copper jacketed bullets were used, so the shooter must decide what he wants to use, and how often he wants to use them. Don't take my word for any of this, ask Heym, the rifle maker their opinion of shooting mono or steel jacketed bullets, and if they would be somewhat harder on the barrels than if softer lead/copper alloy jackets were used instead.