A new paradigm of mechanism design, called mechanism design over social networks, investigates agents’ incentives to diffuse the information of mechanisms to their followers over social networks. In this paper we consider it for two-sided matching, where the agents on one side, say students, are distributed over social networks and thus are not fully observable to the mechanism designer, while the agents on the other side, say colleges, are known a priori. The main purpose of this paper is to clarify the existence of mechanisms that satisfy several properties that are classified into four criteria: incentive constraints, efficiency constraints, stability constraints, and fairness constraints. We proposed three mechanisms and showed that no mechanism is better than these mechanisms, i.e., they are in the Pareto frontier according to the set of properties defined in this paper.