Stand-alone perception systems in autonomous driving suffer from limited sensing ranges and occlusions at extended distances, potentially resulting in catastrophic outcomes. To address this issue, collaborative perception is envisioned to improve perceptual accuracy by using vehicle-to-everything (V2X) communication to enable collaboration among connected and autonomous vehicles and roadside units. However, due to limited communication resources, it is impractical for all units to transmit sensing data such as point clouds or high-definition video. As a result, it is essential to optimize the scheduling of communication links to ensure efficient spectrum utilization for the exchange of perceptual data. In this work, we propose a deep reinforcement learning-based V2X user scheduling algorithm for collaborative perception. Given the challenges in acquiring perceptual labels, we reformulate the conventional label-dependent objective into a label-free goal, based on characteristics of 3D object detection. Incorporating both channel state information (CSI) and semantic information, we develop a double deep Q-Network (DDQN)-based user scheduling framework for collaborative perception, named SchedCP. Simulation results verify the effectiveness and robustness of SchedCP compared with traditional V2X scheduling methods. Finally, we present a case study to illustrate how our proposed algorithm adaptively modifies the scheduling decisions by taking both instantaneous CSI and perceptual semantics into account.