Regrasping a suture needle is an important process in suturing, and previous study has shown that it takes on average 7.4s before the needle is thrown again. To bring efficiency into suturing, prior work either designs a task-specific mechanism or guides the gripper toward some specific pick-up point for proper grasping of a needle. Yet, these methods are usually not deployable when the working space is changed. These prior efforts highlight the need for more efficient regrasping and more generalizability of a proposed method. Therefore, in this work, we present rapid trajectory generation for bimanual needle regrasping via reinforcement learning (RL). Demonstrations from a sampling-based motion planning algorithm is incorporated to speed up the learning. In addition, we propose the ego-centric state and action spaces for this bimanual planning problem, where the reference frames are on the end-effectors instead of some fixed frame. Thus, the learned policy can be directly applied to any robot configuration and even to different robot arms. Our experiments in simulation show that the success rate of a single pass is 97%, and the planning time is 0.0212s on average, which outperforms other widely used motion planning algorithms. For the real-world experiments, the success rate is 73.3% if the needle pose is reconstructed from an RGB image, with a planning time of 0.0846s and a run time of 5.1454s. If the needle pose is known beforehand, the success rate becomes 90.5%, with a planning time of 0.0807s and a run time of 2.8801s.