Pedestrian trajectory prediction is essential for self-driving vehicles, social robots, and intelligent monitoring applications. Diverse trajectory data is critical for high-accuracy trajectory prediction. However, the trajectory data is captured in scattered scenes, which can cause the problem of data island. Furthermore, artificial aggregation of trajectory data suffers from the risk of data leakage, ignoring the rule of privacy protection. We propose a multi-scene federated trajectory prediction (Fed-TP) method to solve the above problems. As our key contribution, a destination-oriented LSTM (Long Short-Term Memory)-based trajectory prediction (DO-TP) network is proposed in each scene to forecast future trajectories in an encoder-decoder manner. The independent training using trajectory data in each scene can prevent data leakage and achieves high privacy security. As another key contribution, a federated learning framework is introduced to break the scene limitation by conducting distributed collaborative training. The performance of different federated learning methods is compared on public datasets, including ETH, UCY, and Stanford Drone Dataset (SDD). Compared with FedAvg and FedProx, FedAtt is more suitable for pedestrian trajectory prediction. Experimental results demonstrate that the proposed method has better data privacy security than directly training on multiple scenes and superior prediction performance than training on a single scene.