Two-sided matching platforms can control and optimize over many aspects of the search for partners. To understand how matching platforms should be designed, we introduce a dynamic two-sided search model with strategic agents who must bear a cost to discover their value for each potential partner and can do so non-simultaneously. We characterize evolutionarily stable stationary equilibria and find that, in many settings, the platform can mitigate wasted search effort by imposing suitable restrictions on agents. In unbalanced markets, the platform should force the short side of the market to initiate contact with potential partners, by disallowing the long side from doing so. This allows the agents on the long side to exercise more choice in equilibrium. When agents are vertically differentiated, the platform can significantly improve welfare even in the limit of vanishing screening costs by forcing the shorter side of the market to propose and by hiding information about the quality of potential partners. Furthermore, a Pareto improvement in welfare is possible in this limit.