Don't carry your alcohol on your person(carry-on bags). This is not allowed. You place beverages in your 'checked' luggage. Checked luggage is the luggage brought on by the baggage handlers at dock side. Any alcohol brought on (by you) 'during' the cruise (ports of call) is collected by security and returned to you at the end of the trip.
I was on the Rhapsody week of 1-5 to12. In Cozumel I bought a large bottle of Kaluha. I had a beach bag with me and just stuck all of my purchases, including the Kaluha, in the bag. The whole thing went through security and nobody said a thing. I was surprised since I expected them to hold it until the end of the cruise. There was a table set up with a sign reading something like 'Alcohol Holding ' but there was no one manning the station. I guess it is hit or miss. Maybe if I came back on board with several bottles in a carton they would have held it. I know that in the East. Carib. the shops deliver the booze to the ship for you and it is sent to your room on the last night of the cruise. There doesn't seem to be that kind of arrangement with the shops in Cozumel. Perhaps the Duty Free store may do that, but not the little places.
I would follow the advice above for bringing anything from home. They may be a little stricter when embarking at the begining of the cruise.