Excellent question! And there are a few different answers, really. The first, easiest one would be applicable to an army on the march. When an army was traipsing around the countryside for whatever reason, their camps would be temporary structures, created and destroyed daily. In this case, a cesspit would be dug and would be used as a latrine. Next day, a new cesspit in a new spot. Digging a hole is relatively basic, simple, and doesn't take too long when you have a few hundred people who are ready and willing to do the work!

The question that I think you're asking, though, is about the larger garrisons that were planted around the Roman empire (specifically the hotspots) right around the time when we start counting years forward instead of backward. That answer is slightly different! Latrine technology in general was actively adopted by the Romans during the second century BC, and since the army was renowned for having the best engineers in the Roman world, it was only natural that they did some fun stuff with that knowledge. Unfortunately for the majority of the soldiers....well, they didn't get those benefits. Even in the permanent camps, their latrines were cesspits lined in wood or clay to prevent leakage into the drinking water.

But! You see the small problem here. When there's no leakage, even forgetting the stench of a gigantic port-a-potty, that waste isn't going anywhere. So there were a couple of solutions to that problem: installing drains and assigning someone to stercus duty. We've actually got one extant document that lists the different details of one unit of soldiers, one of whom got that aforementioned lucky assignment.

But! The Romans also brought class to their toilets. The officers, at least, had their own latrine area set up nearby the bathing complex. That sounds a bit odd on the surface, but then you realize that they could flush. It was a relatively simple matter to divert some of the water that was used for the bath houses to flush the waste out and into the nearest river.

Hope that covered it! Feel free to ask if you have any more questions. If you'd like to read some more on Roman bathroom stuff, I recommend checking out The archaeology of sanitation in Roman Italy: toilets, sewers, and water systems. The Google Books preview is unfortunately pretty restrictive, but I can confirm that the book itself does, in fact, have a chapter specifically on the Roman army :)