I have to second Bob's remarks. Alligator snappers are a very long lived, and if cared for properly will get huge. Not planning for the long term care of the animal is simply irresponsible. A properly cared for, and properly housed alligator snapper isn't just a pet, it's a potential heirloom.
Sure babies are cute, but not planning is what ends up with so many reptiles abandoned to shelters, or simply dying from neglect. Alligator snappers are simply not a pet for just anyone who wants a "cool" turtle.
That being said, the general rule with most animals, is to not mix species. Differing care needs, diets, and health concerns make it just not a good idea. Then, if you have multiple animals in an enclosure, and one gets sick, you're pretty much guaranteed to get its cage mate sick. Differing species can have different susceptibility to diseases. Yes, zoos often do it all the time, but they also have a dedicated staff of keepers and veterinarians to actively monitor all animals, and often have large (well, at least larger than almost any private keeper's), naturalistic habitats which often is enough to keep animal interaction to only incidental encounters.
Other turtles are most certainly on the diet of large snappers, they are generalists and will eat pretty much anything they can catch. They aren't likely to try and eat anything that doesn't fit in their mouth, but their jaws are incredibly strong and a bite can do some serious damage even if they aren't actively trying to eat another turtle. If you've ever seen a snapper rip apart a fish with a single massive chomp, it's pretty amazing. They actually have a pretty calm disposition for the most part, and are not typically prone to biting unless harassed. They have a rather undeserved reputation for being big and mean.
The alligator snapping turtle is listed as a "threatened species" and is protected from collection through much of it's native range, though it's frequently farm bred for the pet trade. The only federal law covering them is CITES, which restricts their export from the US. State laws would be the main restricting factor in keeping them, but one should also look at their city bylaws. Many cities have restrictions on keeping species which can be considered dangerous. Of course, they leave the determination of exactly what is dangerous up to the local animal control, so you should call their office to make sure snapping turtles aren't on their list. Just because they're sold in the city doesn't mean they're legal to keep as pets either.