For solo camping, I usually carry it out in a wee plastic bag. If it's a well used area where ground contamination isn't a problem, I dig a hole and bury it, and burn the bog paper. For a communal camp, or a long term base camp, you can't beat a thunder box. That can be a porta potty, or an inverted wooden box with a hole cut in the lid, with a plastic bucket inside to catch the jobbys. A screen round the thunder box is useful as well for a bit of privacy. The bog roll can be kept on a pole stuck in the ground. A good way to protect it from the rain is to take a five litre plastic container, cut a square hole in the side, pop the bog roll inside and slide it and the bog roll over the stick. Then when you want to peel off a bit paper, you just put your hand in the hole and pull sufficient paper out to wipe your bits. The container covers the roll and will shed rainwater because it's upside down.
Eric