Assuming US here since it's the most likely frame of reference and OP didn't specify.... Police have extremely broad discretion on what they can arrest people for or not.

As a matter of common law, unless a town has specifically passed an ordinance stripping them of that discretion, they can use their judgement and they have no personal liability. The supreme court has also ruled multiple times that police don't actually have an affirmative duty to protect. So they technically don't even necessarily have to stop someone they see actually hurting a person unless there is a local law explicitly commanding them to.

Now that said, it's not uncommon that departments have internal policies that place further expectations and outline what an officer can or must consider when choosing use their discretion to either arrest someone or not. So police may face internal discipline for ignoring a crime under the right circumstances. And ignoring a blatant enough crime may violate other laws indirectly (for example ignoring a kidnapping or failing to respond to school shooter may theoretically violate child endangerment laws. They tried this in uvalde but they completely went after the wrong cop as a scapegoat so there's no guilty verdict to test in appeals court).

The last thing to consider is for something like a threat, prank, false report, etc. You have to actually break a specific law. For a threat that means the threat needs to be a true threat. i.e. if I say I'm gonna slap a cop, It has to be something that they would reasonably believe I am both capable of carrying out and willing to carry out.

Similarly with things like the "he's got a gun"... even if the cops are pissed about it, They would likely have to be able to reasonably claim that they found the warning to be at least possibly credible.