The simple answer is, "no". They don't have to tell the consumer anything, and in my experience, never do.
Although I'd point out that they're supposed to cease all collections efforts of all kinds. If they send some kind of a record to a credit bureau subsequent to the date the purported debtor mailed a timely demand for verification letter, I'd say that is an "attempt to collect", and would subject them to a lawsuit.
The fdcpa is what's called a "private attorney general" statute, in that it shifts law enforcement from the government directly to the affected citizen and, in theory, gives him an incentive in the form of enhanced statutory damages and attorneys' fee. In theory, that's a way of both getting the remedy closer to the harm and shifting the burden to the person who's in the best position to determine whether it's worth taking action.
That's all in theory, of course, because the people who are most affected can't go out and hire a lawyer, and don't know what to do about such things. But the burden is on the consumer to take action. The FTC and state agencies can enforce the law, but they don't have the money or the staff to do so effectively. (That's why I like this website, because it's getting information to the people who need it.)