For most winter pursuits, mittens don’t need to be fully waterproof, but they should be water resistant.
A water-resistance mitt will hold up for most snowy days, even if you are really getting your hands into that snow as you work or play.
However, if you are heading out on a super wet powder day in a damp climate, a fully waterproof mitten might be a better option, like designs with a GORE-TEX membrane, to ensure your hands stay completely dry and warm.