I think it all depends upon your teens....you are better to judge them as their parent.
Our daughter, who was 10, went on our last cruise in March on the Carnival Inspiration. We gave her a walkie talkie to use and let her explore the ship, enjoy the kids programs, etc...and told her she had to be back at certain times (ie for dinner, if she wanted to see a show, etc...). If she wasn't back when specified then we would have kept a really close watch on her but she proved to be very responsible for her age.....She even met a friend in the kids program and they hung out alot! Of course on port days we did excursions together and we never left her on the ship alone. She always knew where we were and vice versa and she actually learned the ships layout very fast!! She had her Camp Carnival listing and she just stayed busy.
On our next cruise (RCL Mariner of the Seas) which is twice the size of the last ship we were on she'll probably be fine too. But I still will have rules and times we'll have to meet or be together. I believe you can have fun and let them explore and do their own thing with limits.
When she gets to be in the teenager I'm sure she'll also want to do her own thing even more...but I'd still want to do the "family" stuff (ie dining together-some shore excursions) also. Afterall it is a family trip we're taking when we go on a cruise anyways and we want to spend alot of time together but also allow her to explore and start getting her independence.....
I don't think I'd ever leave her alone during a shore excursion (shore excursions and dining were basically our only family time!) because I wouldn't want anything to happen to her when I'm not there and I wouldn't know if I'm off the ship. Guess that is just the over protective Mom in me.