Hiya!

You're asking the right question - and in the case of Simo Häyhä, the answer is an emphatic 'not at all.' Methods for confirming a sniper's 'kill-count' have varied widely by time and institution, so I'll address the matter of Simo Häyhä directly, since the Internet's ongoing obsession with him is a particular bugbear of mine - one which I've discussed here and here.

Simo Häyhä's 'confirmed kill-count' -a 'confirmed' count that seems to vary between ~500 and ~700 depending on who you ask, should not be trusted because it's a fabricated and inflated estimate. Indeed, Simo Häyhä wasn't even a sniper, but a marksman and infantry section leader. By his own account, Häyhä was not attempting to keep count himself - Häyhä and his fellow soldiers (once again, he was an infantry section leader - a similar role to a corporal, and not a sniper) spitballed most of their figures, and even the original estimates of his 'kills' are vague at best. More realistic estimates figure in at ~'kills,' however no scholarly English language discussions of Häyhä's activities exist. Unfortunately, a number of very poor pop-history books do discuss Häyhä, inevitably repeating the same tired myths I've outlined above.

For a more detailed discussion of Häyhä and the circumstances of his activities, as well as the discussions surrounding him, I'd recommend you read /u/Mosinista's excellent post here from a recent /r/History thread. If you have any other questions, I'll of course do my best to answer them!