Nope, if it's reported to them, they do tend to follow up on it. A couple of anecdotes worth noting here.

Long, long time ago I gamed with a guy who worked in membership for the NRA. He was on the VRE (commuter train) and the guy next to him struck up a conversation including what DC-folk usually ask, "what do you do for a living?" NRA guy answers with where he works and the other guy starts ranting about how much he hates President Clinton, and how he "wished someone would just shoot him". NRA guy said basically, look I don't like him either, but that's not cool, man, and the conversation ended. The next day, the Secret Service showed up at the NRA to talk to the guy I gamed with. Apparently someone had overheard and called in a report. It wasn't a huge thing, they interviewed him for an hour or two about what he remembered about the guy on the train, and said that even if it's probably just someone spouting off, they still have to follow up on it if it's reported to them.

I also knew someone who volunteered sorting mail at the White House. The sorters get instructions on how to deal with negative mail that basically amounts to if they say they hate the President and think he's a jerk or is going to burn in Hell or should be impeached, that gets put in one pile, but if it mentions violence in any way (even if it's "I hope you die in a fire, asshole"), it gets put in a bin to be sent to the Secret Service for further evaluation.

They do follow up on "random yutz spouting off" reports and letters, even if it's to do a cursory investigation and decide the person's harmless.