If you talk to the people who make the primers, the usual answer you get is that the magnum primers contain a bit more priming compound than the standard primers.
This produces a slightly more "powerful" flash, possibly of slightly longer duration. With some combinations of cartridge case and powder this can raise pressures but how much is not a standard thing and I have seen data ranging from "no appreciable difference" to 5000psi difference.
IF your not working with absolute max permissible pressure, you can PROBABLY swap standard for magnum primers without any issues.
HOWEVER, the RIGHT thing to do, to ensure not just safety but also acceptable performance is to drop the load back to starting levels and work up using magnum primers.
Just remember every cartridge component combination could give uncommon or unexpected results. Probably won't, but could.
I have heard stories about switching to magnum primers in small cases (such as 9mm pistol) and having the "extra" force from the magnum primer unseat the bullet prematurely. I have no idea if this happens or is just one of those stories, but it seems possible, under certain conditions.
Work up your loads using the usual methods, I doubt you'll have any problems.