Out-of-stock (OOS) detection is a very important retail verification process that aims to infer the unavailability of products in their designated areas on the shelf. In this paper, we introduce OOS-DSD, a novel deep learning-based method that advances OOS detection through auxiliary learning. In particular, we extend a well-established YOLOv8 object detection architecture with additional convolutional branches to simultaneously detect OOS, segment products, and estimate scene depth. While OOS detection and product segmentation branches are trained using ground truth data, the depth estimation branch is trained using pseudo-labeled annotations produced by the state-of-the-art (SOTA) depth estimation model Depth Anything V2. Furthermore, since the aforementioned pseudo-labeled depth estimates display relative depth, we propose an appropriate depth normalization procedure that stabilizes the training process. The experimental results show that the proposed method surpassed the performance of the SOTA OOS detection methods by 1.8% of the mean average precision (mAP). In addition, ablation studies confirm the effectiveness of auxiliary learning and the proposed depth normalization procedure, with the former increasing mAP by 3.7% and the latter by 4.2%.