In the presence of self-interested parties, mechanism designers typically aim to achieve their goals (or social-choice functions) in an equilibrium. In this paper, we study the cost of such equilibrium requirements in terms of communication, a problem that was recently raised by Fadel and Segal . While a certain amount of information x needs to be communicated just for computing the outcome of a certain social-choice function, an additional amount of communication may be required for computing the equilibrium-supporting prices (even if such prices are known to exist). Our main result shows that the total communication needed for this task can be greater than x by a factor linear in the number of players n, i.e., n · x. This is the first known lower bound for this problem. In fact, we show that this result holds even in single-parameter domains (under the common assumption that losing players pay zero). On the positive side, we show that certain classic economic objectives, namely, single-item auctions and public-good mechanisms, only entail a small overhead. Finally, we explore the communication overhead in welfare-maximization domains, and initiate the study of the overhead of computing payments that lie in the core of coalitional games.