The coaches know, because as TPM said, they clear the player through admissions before making the offer.
I had this same question regarding my son. On his official visit, which was after he had committed but before he had applied for admission, I asked the HC if it was a done deal or should we wait for admission acceptance before telling family and friends.
He laughed and said "If your kid gets rejected I'm in a world of hurt because he has the highest numbers of anyone in the recruiting class. You can tell your friends."
I later learned that the way it is handled at Duke is that the coach has 9 spots each year for which he can essentially give the nod as long as the player is somewhere in the realm of being qualified on paper (such as at least in the lowest quintile SAT and GPA.)