Assassins will eat other snails of any type, except for really large snails. They'll wipe out MTS, though the biggest ones seem to be more than they can manage. Nerites oddly fall prey to them pretty readily. They can and will also eat shrimp, insect larvae, fish food, detritus, algae, fish eggs, fry, etc. They're omnivores, and quite robust.
They do breed quite prolifically in freshwater.
Personally, I think they may be the next big ecological problem. They breed rapidly, and they're incredibly adaptable and hardy. They also have very, very small eggs and offspring which can hide in just about anything, readily making it down the drain. They can also handle drying, freezing, any water conditions.... I think it is a matter of time before they invade freshwater bodies. We do not have a native, predatory snail in most bodies of water. They will decimate native snail populations, as well as that of aquatic insects, which will have ramifications up and down the food web.
I've found them in tanks that I know I have not put assassins in, simply because they hitched on something.
For those of you who have assassin snails, PLEASE be careful with them. Make sure you are not putting them down the drain or into the trash or compost.