I would say to simplify, and I am by no means an expert, but if they share any type of flesh (coenechyme) then some sharing of nutrients may be occurring. We are talking a myriad of types of corals and different growth patterns here, even differentiations within species, so I'd imagine it would be hard to give a generic answer other than... if they share flesh, they're probably sharing nutrients too.
I have witnessed the same colony of euphyllia have flesh connected throughout the whole colony (during periods of apparent good health) and not connected (during periods of poorer health). The interesting thing is that once the flesh is separated, during times of greater health, I have seen the flesh regrow between stalks.
It would be an interesting experiment for someone to totally encapsulate a connected stalk of euphyllia to prevent it from getting any light and feed adjacent stalks that are still connected with flesh to see if it continues to thrive in a completely dark environment. I would imagine it would be clear from the death of all the photosynthetic zooxanthellae that resides within it but if the polyps are still fleshy and healthy looking, then it would seem that the nutrients shared through the flesh is sustaining it. The hard part would be to figure out a way to isolate the stalk in total darkness without depriving it of the benefits of water motion to be able to supply it with oxygen, remove wastes, and supply nutrients.
-Ty