In many cases the private tours are not much different from those offered by the ship. In both cases local tour companies are used. It's just that one has a contract with the cruise line and the other doesn't.
Tulum is a fairly long trek from Cozumel, and not something I would recommend doing on your own with a private tour. As the others have said, you have "insurance" when booking with the ship. They will wait for you if your tour is late returning - and I have seen this happen more than once. I have also seen passengers who missed the ship in Cozumel waiting on the pier when we docked at our next port. They did not have a pleasant time during the night and previous evening.
Now when it comes to Dunn's River, I think you would be okay doing it on your own. The falls are just a couple miles from the piers in Ocho Rios. But you must exercise caution in dealing with the tour operators in Jamaica. If you want to do it on your own, stick with those offering tours who are inside the fenced off area at the pier. You can encounter some less than scrupulous people if you wait until you get to the main street outside the pier area.