Shape incompleteness is a common issue in point clouds acquired by depth sensors. Point cloud completion aims to restore partial point clouds to their complete form. However, most existing point cloud completion methods rely on complete point clouds or multi-view information of the same object during training, which is not practical for real-world scenarios with high information acquisition costs. To overcome the above limitation, a self-supervised point cloud completion (SPC) method is proposed, which uses the training set consisting of only a single partial point cloud for each object. Specifically, an autoencoder-like network architecture that includes a two-step strategy is developed. First, a compression-reconstruction strategy is proposed to enable the network to learn the representation of complete point clouds from existing knowledge. Then, considering the potential problem of overfitting in self-supervised training, a global enhancement strategy is further designed to maintain the positional coherence of predicted points. Comprehensive experiments are conducted on the ScanNet, MatterPort3D, KITTI, and ShapeNet datasets. On real-world datasets, the unidirectional Chamfer distance (UCD) and the unidirectional Hausdorff distance (UHD) of the method are reduced by an average of 2.3 and 2.4, respectively, compared to the state-of-the-art method. In addition to its excellent completion capabilities, the proposed method has a positive impact on downstream tasks. In point cloud classification, applying the proposed method improves classification accuracy by an average of 14 %. Extensive experimental results demonstrate that the proposed SPC has a high practical value.