You don't alert the press until you're pretty damn sure, though. Military bases can already have stressed relationships with their communities and they don't need to be starting a panic over every change in the FPCON.
This is really the correct point of view for this. In the military, being horribly inconvenienced and having your plans ruined is normal life. We often had unplanned drills of various types in Japan, with no forewarning. You do your job, wait a while, then they announce the all clear and you go back to life.
However, the one or two times people messed up and alerted the local Japanese government, it was a HUGE deal. Heads rolled, people got punished, and apologies had to be written.
That's absolutely correct in my experience. The cover-ups extend to things like accidents involving nuclear weapons, of which there have been many.
It's fair to say that the host countries play along in most cases. If there's no public disclosure of problems the host country government does not have to do anything about them.
Or you potentially decided that the threshold for warning the base is much lower. Because in the worst case, you've just had an unannounced missile drill that's good for response times anyways. If you're on a military base, having a few such alarms per year seems like reasonable training.
If you do this to civilians, by the second alarm of this sort they'll start figuring out how to make the alarms stop.