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).

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.

Checking in with my passport and boarding card when I get to the gate counter has always worked, but yes, they will page you to visit the gate counter agent prior to starting boarding if you don't do it yourself. On the other hand, if my boarding pass doesn't have either statement (other airlines) I've never had a problem with just approaching the gate agent and asking "do you need to check my passport?" - but do be sensitive to how busy they are.