In this paper, we investigate a multi-receiver communication system enabled by movable antennas (MAs). Specifically, the transmit beamforming and the double-side antenna movement at the transceiver are jointly designed to maximize the sum-rate of all receivers under imperfect channel state information (CSI). Since the formulated problem is non-convex with highly coupled variables, conventional optimization methods cannot solve it efficiently. To address these challenges, an effective learning-based algorithm is proposed, namely heterogeneous multi-agent deep deterministic policy gradient (MADDPG), which incorporates two agents to learn policies for beamforming and movement of MAs, respectively. Based on the offline learning under numerous imperfect CSI, the proposed heterogeneous MADDPG can output the solutions for transmit beamforming and antenna movement in real time. Simulation results validate the effectiveness of the proposed algorithm, and the MA can significantly improve the sum-rate performance of multiple receivers compared to other benchmark schemes.