Intersection management with mixed cooperative and non-cooperative vehicles is crucial in next-generation transportation systems. For fully non-cooperative systems, a minimax scheduling framework was established, while it is inefficient in mixed systems as the benefit of cooperation is not exploited. This paper focuses on the efficient scheduling in mixed systems and proposes a two-stage decision framework that makes full use of the cooperation. First, a long-horizon self-organization policy is developed to optimize the passing order of cooperative vehicles in a distributed manner, which is proved convergent when inbound roads are sufficiently long. Then a short-horizon trajectory planning policy is proposed to improve the efficiency when an ego-vehicle faces both cooperative and non-cooperative vehicles, and its safety and efficiency are theoretically validated. Moreover, simulation results verify that the proposed policies can effectively reduce the scheduling cost and improve the throughput for cooperative vehicles.