The rules for urine are very simple -- piss in the river. It is ok to use the wet sand next to the river, because fluctuating river levels would quickly flush it out. Never piss on dry ground or rock -- it quickly dries, leaving a smelly residue that will be wonderfully preserved in the dry desert air.
Handling shit is a much more complex problem -- a significant fraction of the cargo capacity of the rafts is devoted to carrying shit. Canyon Explorations used approved war surplus ammo boxes that hold about 1 cubic foot, and in their experience, for an average size group, one box will last about one day -- 14 boxes for a two week trip. Fortunately, the ammo boxes are airtight, and, with chemical treatment with a little dry bleach powder, there is absloutely no odor.
However, there is a real health risk associated with carrying a large cargo of shit -- and experienced guides take it very seriously. Two dozen people with dysentery in the remote reaches of the Grand Canyon would be unpleasant to contemplate.
All this unpleasant detail has evolved to handle a very real problem: Good group campsites are rare in large sections of the river corridor, and these few get heavy use -- perhaps a group of 20 people every day all summer long. It doesn't take much imagination to realize that all those cubic feet of shit would quickly make formidable pile. Unlike urine, shit is not sterile -- you can't just dump it in the river. Even if it wanted to, the Park Service doesn't have the resources to haul it all out of the Canyon. So those who produce it have the responsibility of hauling it out. It is only fair, after all.