Size: They range from a pound, to several hundred, and mere inches long to big enough to swallow people whole.
Appearance: Mostly excessively large, toothy fish.
Threat: Low to mid. Some freshwater fish could certainly post a threat to an average adult human, but they're generally unlikely to attack without being cornered or otherwise provoked, such as being on the end of a fishing line. And even though some stories report people just vanishing under the surface, most are just going to give you a nasty bite.
While lake monsters are actually fairly common, river monsters are pretty rare. This can easily be attributed to the fact that rivers aren't closed systems. They dump into lakes, larger rivers, and eventually the sea. So many river monsters may simply be temporary visitors, and only swimming through the rivers for short periods.
One of the most well-known river monsters was the White River Monster, which has been reported infrequently for almost 50 years. The creature has even claimed to have had an effect on the American Civil War, being blamed for the overturning of a boat that was carrying supplies.
Many reports of monsters in rivers are just tales of fish that are much bigger than any that people have previously seen or caught. "The one that got away" so to speak.
It's also worth noting that the shark attacks that inspired the movie JAWS actually happened miles upstream from the ocean, and at least one species of shark, the Bull shark, is known to be able to swim inland.
Of course, there are always a few unexplained deaths near water. Like I said, human beings are not aquatic by nature. Many deaths are likely due to people slipping under, drowning, and their bodies getting washed downstream or just getting caught among roots, the stories of river monsters being created in a way to have something to blame. There are certainly stories enough of fish that can do some significant damage with their bite, and it's not hard to believe that some species could kill and eat people if grown large enough.
Unfortunately, there's really not much in the way of books or movies regarding river monsters. Most focus on lakes or the ocean. The Animal Planet series Lost Tapes has an episode about the White River Monster. Also, although not focused on cryptids, the series River Monsters includes several stories of super-sized fish and legends of giants that feed on human flesh.
Seriously, check this show out!
So, if you go out on the river this summer, be careful. You might have more on the end of your line than you're expecting.