Most righties actually do. The most famous example, for us Finns at least, is this Teemu guy... the last name escapes me at the moment. (Incidentally, Pulkkinen does too.) Of the names who have lately represented Finland, I should also mention Donskoi and JMA. Other notable examples include Mitch Marner, Bobby Ryan, David Pastrnak.
Of the opposite examples, I can't really tell off the top of my head who dig the left wing besides Laine and Ovie.
The reason for this is that it's easier to receive a pass on the move when you're on the wing that responds to your handedness. Though it's easier to one-time when you're on the opposite wing. So the most common thing you see is a righty on the right on ES, but on the left during PP. Again, the other Teemu besides Pulkkinen is/was a prime example.