There are books about how to environmentally deal with your own waste in the woods, there are blog posts, and then there are frustrations as you come across places where people haven't buried their waste or left tp everywhere (by the way tp does not break down quickly in the woods - bury it or take it with you.)
Due to these frustrations and a number of other reasons I actually like outhouses. (Many of the folks I hiked with didn't and would avoid them.) Then again, I grew up with one at the cottage where we spent our summers (and every other weekend we could). I aimed to stay at shelters in part because of privies. (I also hate carrying a wet tent). By the way not all the shelters have them. On my walk to Maine I found a few in PA that didn't have them - rather disappointing - especially since trying to dig down deep enough to properly bury waste is difficult when everything is rocks.
There are a variety of styles on the AT each operated by a local crew. Some would like you to pee in the woods and in others where liquid is wanted. Some where you should check that there is leaf debris in the outhouse before grabbing your tp and heading to 'go,' and a few where they didn't smell and tp was provided. It didn't matter to me. I planned my days with the hope of starting near one just made the start to my day simpler.
Below are few pictures of the variety available as you hike. Including the Moose Mountain Shelter in New Hampshire which surprised me with being rather 'airy' and a lovely view of the tents. (It was the only privy I came across written up many times in the shelter log.)
Whatever you do, take care of your poo.