Two-sided matching platforms, such as those for labor, accommodation, dating, and taxi hailing, can control and optimize over many aspects of the search for partners. To understand how the search for partners should be designed, we consider a dynamic model of search by strategic agents with costly discovery of pair-specific match value. We find that in many settings, the platform can mitigate wasteful competition in partner search via restricting what agents can see/do. Surprisingly, simple restrictions can improve social welfare even when screening costs are small, and agents on each side are ex-ante homogeneous. In asymmetric markets where agents on one side have a tendency to be more selective (due to smaller screening costs or greater market power), the platform should force the more selective side of the market to reach out first, by explicitly disallowing the less selective side from doing so. This allows the agents on the less selective side to exercise more choice in equilibrium. When agents are vertically differentiated, forcing one side of the market to propose results in a significant increase in welfare even in the limit of vanishing screening costs. Furthermore, a Pareto improvement in welfare is possible in this limit; the weakest agents can be helped without hurting other agents. In addition, in this setting the platform can further boost welfare by hiding quality information.