I understand that the implementation of feces and toilets is already on the Eternal Suggestion Voting thread, but how exactly it is to be implemented isn't explained.
I thus propose the following system:
Poop:
Just like a dwarf needs to eat or drink, a dwarf will need to poop. Excrement will be handled as an item and cannot be traded.
When a fortress is abandoned, EXCREMENT WILL STAY IN THE EXACT SAME PLACE. The last thing anyone wants is turds strewn all over the fortress for no adequately explained reason.
Latrines:
A latrine will have to be built over an open space, and have a room designated from it(pooping in public? Hell no). When used by a dwarf, excrement will be deposited on the tile below the latrine.
In a normal latrine, the feces will eventually rot away by itself, generating miasma(disgusting the unlucky dwarf inside). In a latrine over magma, the feces will simply burn away. In a latrine over water, the feces will dissapear, turning the water into sewage water the same way blood disperses in water(feces will not dissapear in water that has already been turned into sewage water, so the water must be kept running).
Chamberpots:
You may also use chamberpots, which are built on top of a normal floor. When a dwarf poops in a chamberpot, excrement will be stored in the chamberpot itself, the same way items are stored in coffins. A chamberpot with 5 or more feces in it will be unusable.
For the purpose of emptying the chamberpot, a dwarf designated with the 'gong farmer' labor will pick up a gong bucket(using the same bucket dwarves use to supply the wounded would be baaaad), and use it to empty the chamberpot, dumping the feces in a designated sewage dump zone(which may be a landfill or an open space over water/magma). The gong farmer will get an unhappy thought from having to dump poop initially, but will get used to it just like how military dwarves get used to death.
No toilet:
Without a toilet, your dwarves will poop outdoors, preferably in a shrub. They will get a minor unhappy thought from this. I would propose that they poop in any open space over water/magma, but there are just too many potential complications.