We study the communication cost (or message complexity) of fundamental distributed symmetry breaking problems, namely, coloring and MIS. While significant progress has been made in understanding and improving the running time of such problems, much less is known about the message complexity of these problems. In fact, all known algorithms need at least $Ω(m)$ communication for these problems, where $m$ is the number of edges in the graph. We address the following question in this paper: can we solve problems such as coloring and MIS using sublinear, i.e., $o(m)$ communication, and if so under what conditions? [See full abstract in pdf]