You can choose to propose either publicly or privately. Even if you choose to propose marriage in public, snowy locations are such that there will be only a few people around you compared to a bustling metropolis.
Such kind of a romantic proposal also throws up some considerations for you to take. The first thing is would want to simply go down on one knee and pop the question, and do it with a diamond ring obviously. Taking the icy rock to an ice mountain setting may be an extremely romantic idea, one that could do the trick for you.
You could even go on a ski trip and proposing her when the right time comes. Apparently, that is the whole idea of proposing someone at offbeat locations – just to make it more casual, romantic, and private. You could even choose just a solitaire presentation setting to make it work, and choose a more preferred setting when you get back home as an engaged couple.