First, let's set the scene. It doesn't matter what era the game's taking place in, for this purpose, as the scene's broadly the same. The characters have done something blatantly illegal, and evidence remains that might indicate who did it, and how. Moreover there's always the chance that the police know more than they ought about the occult aspects of the campaign, whether it's infested with bloodsuckers, or a mere bubble of fragile humanity lodged in a sea of unthinking chaos. Then what happens, and what ought the characters to do about it?
The Keeper needs to bear in mind that:
- The cops don't know everything. Outside of television and Agatha Christie, there really isn't a master detective out there who, having taken only a cursory look at the crime scene, understands exactly how it was done and who did it. To the players, what happened may seem painfully obvious. To the first responders, it's a pool of blood, some expended rounds, and maybe some body parts or a corpse.
- Finding out what happened takes time. Investigating the scene, making drawings or taking photographs, collecting evidence, the canvass, interviews;all these things take hours, days, maybe even months before any kind of resolution is achieved.
- There are always other suspects. That cult leader or necromancer the characters put down like a rabid dog undoubtedly had plenty of enemies, and the police are going to go through that list name by name. It only gets more complicated if anything screwy's on the scene, like a burial pit with a dozen or more other dead bodies in it, or physical evidence that suggests the decedent is someone, or thing, completely out of the ordinary.
- There are always other cases. To the protagonists, this is the only crime in town, but to the cops this is maybe the third shooting tonight. It could just go cold, as so many cases do, or something much more urgent - a terror threat, a spectacular burglary - may come up to distract the law.
But what happens when the police know more than the characters would like them to? Some fantasy novels - Ben Aaronovitch's Peter Grant series, for instance - assume that there are organizations, or people, within the police who understand the spooky stuff, and who can take action against it. What happens then?
A great deal will depend on the setting. Trail of Cthulhu posits two types of game, Purist, and Pulp. Bookhounds of London further divides into Arabesque, Technicolor, and Sordid. A Pulp or Technicolor game is probably best suited to a campaign setting in which the police have a Peter Grant style organization within it, dedicated to dealing with occult threats and capable of using arcane powers. In that kind of setting the police could be very useful allies, but even more dangerous enemies. If the Occult Sciences secret division of whatever passes for Military Intelligence decides that the player characters know too much, it has all the resources of a vengeful Government to bring to bear on them. At that point the characters had better leave the area as soon as they're able.
A Purist setting allows for more nuance. There probably are policemen who've seen too much; there may even be one or two who've begun putting the pieces together. But those are lone wolves, disbelieved - even mocked - by their fellow officers. Promotion doesn't come easily to them. In some situations - an Arabesque campaign, for instance - these policemen may have other resources to bring to bear. Imagine a setting in which the constable who always seemed to have that certain something about him, is actually a Knight of the City, protected by London itself; though he, oblivious, may not realize it, the City is helping him each day, keeping him from harm, helping him in his work. Or one in which, being a seventh son of a seventh son, he has the gift of second sight.
Sordid has its own complications. There may be police who know a little about the occult, or even the Mythos, but nothing's free in a Sordid world, and everyone's on the take. Expecting someone like that to be a faithful ally is expecting far too much. On the other hand, a cop with that kind of knowledge may throw their weight around, leaning on the characters when they need to, even blackmailing them if the opportunity arises.
Night's Black Agents is a different situation. In those games, the police exist mainly as roadblocks or barriers, to be thrown in whenever Heat gets a little too much for the characters. Exciting chase scenes can be had, maybe even a gun battle or two, but the characters generally aren't expected to do much interacting with the local law on a face-to-face basis, particularly since the agents probably aren't going to be in one place long enough. Athens tonight, Paris tomorrow, and if they happen to leave two bodies behind in their travels, they're too far away from the scene to care.
In those situations, it may be worth the Keeper making a note of where the characters have been, in case they go back. Say they made a mess in Athens, went to Paris, had several more adventures, and then, after the Athens debacle has long since faded in the players' memories, they go back to Greece - Heraklion, why not - for another piece of the puzzle. The characters may have forgotten, but the Greek police won't, and old sins may come back to haunt the agents. Sure, they've changed covers since then, but it only takes one careless moment for Heat to suddenly skyrocket, as the cops realize the doofus they caught taking photographs of a military base is actually the same person involved in multiple homicides a year ago.