If someone is training to hunt monsters in a modern day setting (like an urban fantasy story), what might be a good starting point for learning how to use weapons? If they don't specialize in a specific type of entity, and it's one that can be brought down physically, what other things do you think it might be good for someone to be taught?
So, I've said this before as a bit, but monster hunting wouldn't really be about combat most of the time. There's probably some edge cases, like vampires, where you're fundamentally looking at a human with superpowers, but against the vast majority of critters, monster hunting would be more like, well, hunting, or pest control.
If the monster is exceedingly lethal, you don't want to be getting into fights with it. That means taking a more methodical approach to identify and exploit weaknesses, and then eliminating it like any other pest.
Ironically, while the book is a bit crude at times, one of the first examples that comes to mind is Vampire$ by John Steakley, (and the film adaptation by John Carpenter.) The priority there is to identify vampire nests during the day, then use specialized stake guns attached to winches to drag the vampires out into the sun, before collapsing the building entirely.
This means, you're probably better off looking at people with backgrounds in animal control, exterminators, park rangers, game wardens, and similar backgrounds, rather than military or law enforcement backgrounds. (Though, technically, rangers may fall under the header of LEOs.)
Effectively dealing with any animal (and most cryptids tend to fall into this category) involves learning about its behavior. Now, obviously, if you're dealing with some incredibly reclusive creature that's evaded detection by the modern world, it does strain credibility a bit when someone wanders in with a complete understanding of their behavior. (See: Every bigfoot expert ever.) But, the basic premise is foundational if your characters are going to go out hunting those creatures.
When you're dealing with more overtly supernatural foes, then the focus on information they'd need would change somewhat. But, in a lot of ways, the same template holds. It's critical for your hunters to understand the limits and favored behaviors of those creatures.
Now, obviously, you can have very different approaches. I was recently reminded of the old World of Darkness Demon Hunter X splat, where one of the groups detailed (in a modern urban fantasy setting) were using extensive cybernetic augmentations to directly fight monsters (with a very anime inspired aesthetic.) If you have the setup for it, you can go in really wild directions with your urban fantasy monster hunters, and the advice of a relatively low-power group that treat cryptids the same way you'd treat a rat infestation might not be applicable. Similarly, hunting vampires and werewolves may involved more conventional, military tactics, if that's what you're looking for.
It does depend on what your story calls for, and I'll readily admit, my main observation here has more to do with how modern horror writing tends to present these creatures, and then looking for a practical solution to the problem they pose. (Also the juxtaposition is funny. You start out expecting Blade, and instead get some random guys in jumpsuits with animal control poles, who are going to grab the horrific deathbeasts, stick them in crates, and ship them off to a preserve in Montana.
-Starke