As I understand it, shield in coax works in a way to trap the electrical field between center wire and inside of the coax, not radiating outwards. Return current due to mismatch will flow on outside.
In order for twisted pair shield to work as you said, I'd think it'll have to be one connector at each end and shield to entrap both center wires at the same time. In this case, I think, the electrical field takes place between the twisted pair. Shield is merely a shield, and it is to be connected only on one side.