Both can occur. I've had a lot of crimped .223 brass that you could still push primers into without removing the crimps. That's rare with .30-06 and .308 military brass, IME, but with the smaller ones it's not so unusual. There is apparently a less severe crimp on a lot of the latter.
A fair amount of foreign brass has pockets that are a little smaller than US standard. 5.85-5.90 mm (0.2081"-0.2083") seems common for large rifle. I bought 1000 pieces of IMI .308 Match brass recently. It all has to go through my 25 year old Dillon 600 swager before I can get the fat and burred Russian TulAmmo and Wolf primer cups into them. I also had 1000 pieces of IMI Match .45 Auto brass I bought one time that also had to go through the Dillon 600. Funny that the fattest and hardest to seat primers and the narrowest primer pockets both come from across the oceans to our east, but there it is.
Swaging is the fastest approach to handling either crimped or undesized pockets. Reaming undersized pockets by hand has alignment issues. The only reamer I have that works really well is the tool for my Wilson trimmer. The problem is that it takes an eternity to do many cases that way, and my fingers are getting pretty raw feeling after about twenty cases. But for a smoothly cut, easy to seat into primer pocket, it's great.
Swaging has to push the extra brass somewhere, and it ends up around the perimeter of the primer pocket like a slight crater berm. If you drag the head of a case whose primer pocket has been swaged over sandpaper that is on a flat surface, you see it mark that pocket perimeter. The tiny berm flattens back down as soon as you fire a round in the swaged case. Other than perhaps acting as a weak headspace shim, it doesn't seem to have any effect I can detect. It may even help with the tightening. The only downside is that I've had a few individual cases where that berm has been big enough that when I fired the round the brass pushed in over the pocket perimeter enough that I had to swage the case a second time before I could seat a primer easily again.