The obvious experiment here is to have two groups of dolphins positioned so that they can hear each other but not see each other; give one the puzzle, and then, later, give the other the same puzzle. If the second group solves it consistently faster than the first one then probably information is being transferred. (Although it could just be that the second group is more motivated after hearing the first group get excited about something.)
Actually arranging an ethical but meaningful experiment with bottlenose dolphins, which are quite big and are highly opinionated, is left as an exercise for the reader...