If you spend many long winter nights in tents, you will eventually have to get up in the night and GO #2. Under ordinary circumstances you just walk out into the snow with your shovel, dig a hole, clean up with some paper or snow, re-dress and go back to your tent. Not a problem right? Been doing that for a long time.
Some people will attempt to do the "blue bag" thing inside their tent - good luck, hope you're camped alone. Men often have a pee bottle, but women generally will have to follow the #2 concept for #1 also.
BUT if there is a blizzard outside you better have a plan before you go out.
Forced to to go out in a blizzard to relieve yourself, you may not have a great deal of time and patience with dressing for the weather, but if you do not dress with a plan for what you intend to do, you could end up half naked in a cold wind and get you precious dry clothes soaked.
Put it this way. You will have to drop your pants/bibs and if they have shoulder straps, you may have to remove your coat to pull them down. Dress so that you pull you lower insulation down without having to open your coat.
Now if you simply pull down your pants to do your business in a blizzard, those pants will be full of snow when you go pull them up, so as you pull them down you must fold the top edges together (easier with bibs) such that only the outer layer of pants can be touched by snow. Likewise when pulling them back up, be careful to not let snow in. Don't worry too much about properly layering your clothes - just hurry back to your tent. Remember when you take those clothes off to get into your sleeping bag, to roll them up in such a way as to keep the insides dry.
Controversy aside, I have returned to areas where we winter camp and noted two things - 1)summer campers do not camp in the same spots as winter campers so they won't see a bit of toilet paper - 2) all I have seen in the summer (after snow melted off) was a few bits of toilet paper from the winter camping.
WARNING - a friend of mine stepped out of his tent in the night and got lost coming back - finally he started yelling and I had to go out and find him. I awoke to hear someone calling for a dog named Jim in the middle of the night. Then it dawned on me - I'm Jim! So I hollered back and got out of my tent and showed my light around in a circle until he hollered "I can see your light" then he came in on his own. He was near the stream and I was scared he had fallen in.
In a white out you may leave a snow cave and never find the entrance again. A cord tied inside to your pack and TIED to you, will help you find your way back. It might be a good idea to hang a small light near your tent or cave for reference.
Its a good thing to have a bit of soap along.