Yes they cope well with quite hard pruning.
I do a large overhanging privet tree pretty hard every other March & it comes back fine.
They are used as hedging along a busy dual carriageway near here. They have been cut back quite severely in the past & it still looks good.
If you're bothered & want to maintain some privacy in the meantime, say do it by a third each time over a couple of yrs. But I'd be inclined to do it all this March & accept that for a while your privacy will have gone, but the resulting thicker, denser hedge will be better for it. J.