Discussion about the importance of communication complexity for MPC?
It's safe to say that any Cryptographer would want his/her protocol to be implemented and used by the public. Ideas in MPC have been floating around for quite some time now and have been proven to be theoretically sound. The challenge now is to make them efficient enough that it is feasible to use them in practice. In the domain of MPC, sizes of the messages blow up very easily in most protocols as you may have already noticed.

Given that in most situations the network speed is a bottle neck compared to local execution times, it becomes very important to reduce communication complexity if one is serious about implementing the protocol.