When I check in online my boarding pass (Delta) usually says "INTL - Verify PASSPORT" at the top if the passport needs checking, or DOCS-OK if not.  They will frequently give me a replacement boarding pass with DOCS-OK once it is checked, but I also have one with a "Checked" stamp across it instead.

However, in my experience I've never needed to go back to the main check-in (outside the TSA area) - in fact on one occasion when I needed to go there anyway they looked at my passport but did not okay it and I still had to have it checked at the gate.  Checking in with it when I get to the gate has always worked, and I have never been paged outside the gate area (but I usually get to the gate with plenty of time anyway).