a lot depends upon what you are planning on doing. If you are going on a tour that involves boat transportation to a dive site for example, i would book with the cruise line just for the peace of mind that if something goes wrong and your boat breaksdown or whatever, the ship will wait for you to return, if you book on your own you are out of luck if you don't get back to ship before she sails. Most land tours you can do on your own with no problem, just allow yourself enough time to get back to the ship You should be on board 1HOUR BEFORE SAILING.
Don't let it scare you - just be sensible. If the ship leaves at 4 and they ask everyone to be back on ship by 3:30, then don't book a tour that ends at 3:00. Leave lots of room. we booked mostly private tours and all was fine. Book the tour at the beginning of the day so if you have time left over you can get yourself back to the ship.
Agree with the allow-lots-of-time rule, since most of the tours we've taken--ship's tours and others--ended half an hour to an hour after they were supposed to. You know Caribbean time! Just give yourself a window for errors and you'll be fine.
That reminds me about our last cruise on the Carnival Triumph and the CD said for us to NOT forget our camera when going ashore....That is so when the ship is pulling out and we're running to catch it we can take pics of it leaving us behind! OMG
We booked all our tours off the ship for our next cruise in March. You just have to make sure the times are right (as some ships switch times and others stay on the time of the port they left from)...and allow an hour at least to make it back to the ship in time.