Semi-supervised polyp segmentation has made significant progress in recent years as a potential solution for computer-assisted treatment. Since depth images can provide extra information other than RGB images to help segment these problematic areas, depth-assisted polyp segmentation has gained much attention. However, the utilization of depth information is still worth studying. The existing RGB-D segmentation methods rely on depth data in the inference stage, limiting their clinical applications. To tackle this problem, we propose a semi-supervised polyp segmentation framework based on the mean teacher architecture. We establish an auxiliary student network with depth images as input in the training stage, and the proposed depth-guided cross-modal mutual learning strategy is adopted to promote the learning of complementary information between different student networks. At the same time, the high-confidence pseudo-labels generated by the auxiliary student network are used to guide the learning of the main student network from different perspectives, and no depth images are required in the inference phase. In addition, we introduce a depth-guided patch augmentation method to improve the model's learning performance in difficult regions of unlabeled polyp images. Experimental results show that our method achieves state-of-the-art performance under different label conditions on five polyp datasets.