It was an old ANBU joke, no one was sure where it came from, or when it started. Of course if you did know, someone would have to kill you so it was a bit of a catch 22. The right answer? You make sure all the ninjas are dead, then go somewhere else just to be safe.
It was a simple truism robbed with humor: there are no secretes.
Asuma wasn't a secretive person, he didn't hide anything, but he didn't brag about anything either. It was no "secrete" who his father was, but no one ever accused favoritism, Asuma got his position by being better then the next ninja and no one could argue that.
It wasn't exactly a "secrete" who he was seeing either, he just, never discussed it. To the best of his knowledge, Kakashi was the first to ever notice. Knowing the silver haired jounin, he wouldn't tell anybody. Still, the next day it seemed like half the jounin's knew. A week later every chuunin knew, and in short order even the genin knew.
How do you keep a secret in a ninja village? The real answer is you don't have one in the first place.