Moss is a very persistent beast, and putting it in a normal composter seems to be no guarantee of killing it and decomposing it.
Like all greens things, strictly speaking it should break down, but many people report that the moss is regrowing a very short time after the compost is spread around your plants. In fact it can appear decomposed but within a day or two it is green and growing again.
Perhaps this is all down to not getting enough heat into the compost, and a high enough temperature might actually kill off the moss. But if you want to avoid the possibility of spreading moss to your currently moss-free garden plants, then maybe throw the moss in the garbage instead of the composter, just to be safe.