I use a bottle. Females have more of a hassle. One way is to make a hole in the snow in the absid and fill it up with snow afterwards. Probably more hygienic than a public bathroom.

A friend told me a story about a snowmobiler in a blizzard. He had to do #1 and was using the very common heavy snow mobile coverall with a hood. He pulled it down and did what he had to do super quick before he pulled it back up again and put the hood back on his head. The next thing happening was a desperate man cutting the hood of and it went flying in the snow storm :-).