Here's what I came up with after thinking on it for a while:

A monster is any non-sentient creature that goes out of its way to menace and attack other creatures in and outside of its ecosystem for reasons that are not natural. Signs that you may be dealing with a monster are:

1. An inordinate number of the creatures kills are not consumed or used in any way.
2. The creature is actively destructive to its ecosystem and/or environment.
3. The creature demonstrates a willingness to go above and beyond what would be expected of a predator in pursuit of food, such as a willingness to scale fortified walls to kill those living within or attacking armed sentients when it is clearly strong enough to hunt for easier prey.
4. Persistence to the point of recklessness.
5. A willingness to incur injury in order to attack others. 
6. A willingness to fight on even having sustained wounds that would cause a normal animal to abandon its hunt.
7. Any behavior that can be construed as "cruel" or "malicious" on the part of the creature where a rational explanation for the behavior can not be found.