Accurate prediction of future trajectories of traffic agents is essential for ensuring safe autonomous driving. However, partially observed trajectories can significantly degrade the performance of even state-of-the-art models. Previous approaches often rely on knowledge distillation to transfer features from fully observed trajectories to partially observed ones. This involves firstly training a fully observed model and then using a distillation process to create the final model. While effective, they require multi-stage training, making the training process very expensive. Moreover, knowledge distillation can lead to a performance degradation of the model. In this paper, we introduce a Target-driven Self-Distillation method (TSD) for motion forecasting. Our method leverages predicted accurate targets to guide the model in making predictions under partial observation conditions. By employing self-distillation, the model learns from the feature distributions of both fully observed and partially observed trajectories during a single end-to-end training process. This enhances the model's ability to predict motion accurately in both fully observed and partially observed scenarios. We evaluate our method on multiple datasets and state-of-the-art motion forecasting models. Extensive experimental results demonstrate that our approach achieves significant performance improvements in both settings. To facilitate further research, we will release our code and model checkpoints.