Recent years have witnessed great success in handling graph-related tasks with Graph Neural Networks (GNNs). Despite their great academic success, Multi-Layer Perceptrons (MLPs) remain the primary workhorse for practical industrial applications. One reason for such an academic-industry gap is the neighborhood-fetching latency incurred by data dependency in GNNs. To reduce their gaps, Graph Knowledge Distillation (GKD) is proposed, usually based on a standard teacher-student architecture, to distill knowledge from a large teacher GNN into a lightweight student GNN or MLP. However, we found in this paper that neither teachers nor GNNs are necessary for graph knowledge distillation. We propose a Teacher-Free Graph Self-Distillation (TGS) framework that does not require any teacher model or GNNs during both training and inference. More importantly, the proposed TGS framework is purely based on MLPs, where structural information is only implicitly used to guide dual knowledge self-distillation between the target node and its neighborhood. As a result, TGS enjoys the benefits of graph topology awareness in training but is free from data dependency in inference. Extensive experiments have shown that the performance of vanilla MLPs can be greatly improved with dual self-distillation, e.g., TGS improves over vanilla MLPs by 15.54% on average and outperforms state-of-the-art GKD algorithms on six real-world datasets. In terms of inference speed, TGS infers 75X-89X faster than existing GNNs and 16X-25X faster than classical inference acceleration methods.