Size: Per the animal seen, though sometimes species are said to be bigger than normal, which is attributed to large amounts of food and a lack of predators or competition.
Threat: Again, per animal. Many of the OOP animals commonly seen though are predators, and/or dangerous to people. This includes large cats and primates.
Indeed, large cats are one of the most commonly reported OOP animals. This includes panthers and cougars in the UK, and lions in the US. Cats are known to be resourceful, and quite good at hiding, so it wouldn't surprise many if they elude capture for quite some time. The infrequency of the sightings would also indicate individuals much of the time, as opposed to breeding populations.
Much of the evidence is through blurry photos, eyewitness sightings, and half-eaten corpses of prey animals. Traps set out catch nothing, and even methods that are tried and true for the species don't seem to work. So the animals remain a mystery.
The other commonly reported OOP animal is primates, specifically chimpanzee's, which have been reported in the Everglades, in New York, and the mountains above Hollywood. (Why isn't THAT surprising?) Keep in mind that chimps may not be as smart as we are, but they are much stronger, and more vicious. They can easily kill a human if the mood strikes them, so are every bit as dangerous as a rogue cougar.
This even extends to the popular Alligators in the sewers myths, where animals have sometimes actually been caught.
Let's not forget the always terrifying Sharks in a narrow hallway.
Movies and books have used this trope quite a bit, as well. Lake Placid is about a crocodile that ends up in New York. Alligator is about a reptile in the sewers of Chicago. It's the basis for the episode Devil Monkey in the series Lost Tapes. Many stories go horribly wrong when they put an animal in an environment it's not supposed to be in. Like when King Kong breaks loose from his bonds in New York city.
So, there you go. Just keep in mind, absence of proof is not proof of absence.