In winter time, things are easy. Don't use toilet paper, make snowballs and use them. Dispose of them by throwing them at your hiking companions. (now some may unerstand my response in the solo thread)
If you're really feeling frisky, wrap the whole 'thnig' you droped, with snow, making a large snowball, and throw that at your friends.
When I climbed Whitney, they gave out 'wag bags" along with the permit. Do your stuff in the bag, then dispose of it after the hike. Unfortunately, I had forgotten about it, which caused an interesting scene at the airport security check.
On a less serious note, we should note that the tremendous increase in winter camping is/will be having an effect on the quality of spring water. It'll take a few more years before we really start seeing the effects.