In today's cosmic ray veto example, you get a coincidence trigger when a muon traverses both counters. It would indeed work for other particles traversing both counters, but most particles are not as penetrating as muons and would only hit one counter.
Cosmic ray muons are quite energetic and most can easily go through two scintillator paddles in a few-meter scale setup. Most particle detector experiments are looking at beta or gamma radiation, or neutrons, etc. which will typically deposit energy over less than a few tens of cm.
Of course what the signal and background look like depend very much on what specifically you are trying to do, and it can sometimes be difficult to tell the signal from the cosmic background. And sometimes it's the cosmic rays that are the signal and the background is from something else... But in pretty much all these cases you use some kind of timed logic to ``trigger'' your experiment, i.e., to count a signal or tell your electronics to record the data.