We took our passports to be stamped ourselves in Nassau, Grand Cayman, and Cozumel.
In Nassau, once you exit the Marketplace building after leaving the ship, go to your left. There is a small security booth...just beyond that is a yellow building with a fence around it and army guards. Just let the guards know you are getting your passport stamped and they will direct you to the proper office in the building. Have done this several times, and each and every time the folks there have been pleasant.
In Grand Cayman, the office is right near where the tenders drop you off and pick you up. Just ask any of the folks who work security in the area and they will direct you to the correct building.
In Cozumel, the building is about five blocks from the pier. Once you exit the pier, you will head to the left. I can't remember exactly where it was, but if you ask any of the security folks working on the pier, they can direct you to the correct building.
Hope this helps!