Who would say had the best snipers in WWI and WWII or during the cold war period?

id say during ww2 it would be probally Germany or Russia.

I'd have to say the Germans. They seemed to have a very agressive and well trained sniper force even early on in both wars. That's just IMHO!

The Russian snipers in WWII had very frightening records -women too.
WWI Germans

Well, I can't say who had the best snipers, but if you look at the number of kills, the highest-scoring sniper was Finnish Simo Häyhä, with over 500 kills.
http://en.wikipedia.org/wiki/Simo_H%C3%A4yh%C3%A4

Yes I'd have to agree with the Finns.
Several texts I have read have mentioned the terrible casualties and neagtive effect on morale Finn snipers caused. This was in reference to the Russian invasion. Apparently their huge adavantage lay in their fieldcraft, local knowledge and unforgiving (of snipers) Russian tactics.

However the Finns where closely followed the Russians whose effective yet simple equipment (in particular their optics) coupled with the trend of encouraging their young folk (males and females) to obtain sharp shooter badges in the pre war years meant they had equal or superior equipment and a trained base of shootists to select from. (However I am not saying a trained shootist will make a good sniper however it is an excellent base from which to select candiates.)

Another interesting point was the that the Russian sniper was lauded as a hero and admired in general by the Russian forces and civilians.
This was quite the oppposite in the Allied forces where snipers, on the whole, where loathed.