Quote Originally Posted by Crocodile View Post
I think you got it right, it means "behave like a rooster". Obviously, the usage could be stretched that way or another depending on the context. In my day-to-day vocabulary that word is used in a sense of acting in a proactively aggresive manner; more specifically, being disproportionally aggressive so that nobody will even remotely contemplate to mess around with them.
Okay, so it's maybe a bit stronger than "to act cocky" in English, though the metaphor is the same.

P.S. I would also suggest "to pound/thump one's chest" (i.e., in the manner of a male gorilla) as a colloquial-sounding equivalent:

Американцы петушатся, что нас обскакали = "The Americans are pounding / thumping their chests over the fact that they'd leapt ahead of us."

However, don't confuse "thump one's chest" (expressing manliness, pride, arrogance) with "beat one's breast" (expressing guilt and humility, as when Catholics say "Mea culpa, mea culpa, mea maxima culpa")