Policing character knowledge (fire w/ trolls) is too much of a hassle, such that I'm willing to handwave it most of the time. As another poster said, it's a good incentive to make your own monsters or remix the ones in the MM.
One of the characters in my party is a dragon sorc, loves dragons, they're all over his background. When he spent 10 downtime days with the scribes at the temple of Bahamut, they created a book of dragons for him full of lore, physiology, etc. While it's in his possession, I let him have access to the stat blocks for dragons from the MM, and he can reroll any failed dragon-related INT check with advantage if he spends 10 minutes studying his book for information on the specific topic. Obviously it really only comes into play against dragons, but a similar item could be made really for any creature type, or even an in-game monster manual, like the Winchester's family journal/diary thing. Not really applicable in combat, but if a party is getting stomped or unable to hurt something, they could fall back and consult something like that for a few minutes before returning to face the creature.
Depending on the setting, with folks like Volo running around and the rise of adventurers, I feel like such a book would sell extremely well and the knowledge therein could be quite common amongst veteran adventurers. YMMV. You could make it rare and award it like a treasure.
For most other creatures, when they first appear, I give information to the party from the outset that I think the characters would have ICly based on backgrounds. I pretty much always make sure at least one character knows a vital weakness if the monster is impossible to defeat without that info.
Insight checks to determine what a creature's tactics or strategy might be, or if he seems to be avoiding or wary of a particular character, likely to flee, or fighting to the death.
Martial characters get automatic insight into whether or not an enemy fighter is better or worse than a typical city guard or recently fought monsters (a more limited version of the Battlemaster ability) unless the enemy is deliberately trying to look weak (in which case deception applies).
I allow characters to make an intelligence check to confirm a specific fact, This has to be a specific fact, like "Do I know if this target would be hurt by heat metal?". I also have begun introducing reaction rolls to speed things up. Basically, if they're about to do something less effective, for instance, casting heat metal on a helmed horror, I ask them to make an int (arcana) (in that case) check and if they score above a certain number (pretty much 10, 15, 20 like others have mentioned) they "remember" at the last minute it's less effective to do that and I'll let them change their action accordingly.
I allow medicine checks to verify more information on creature health, if it makes sense for the creature (I.E. not undead). Aside from "bloodied" this is the only way they can really figure out HP info. Additionally, I may give information on a living creature's physical characteristics based on observing physiology, animal handling or nature if its a beast or monstrous type creature.
I give a free perception check to realize that a monster isn't as hurt by an attack as you would expect (resistant), or particularly vulnerable.