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With the wider adoption of edge computing services, intelligent edge devices, and high-speed V2X communication, compute-intensive tasks for autonomous vehicles, such as object detection using camera, LiDAR, and/or radar data, can be partially offloaded to road-side edge servers. However, data privacy becomes a major concern for vehicular edge computing, as sensitive sensor data from vehicles can be observed and used by edge servers. We aim to address the privacy problem by protecting both vehicles’ sensor data and the detection results. In this paper, we present vehicle–edge cooperative deep-learning networks with privacy protection for object-detection tasks, named vePOD for short. In vePOD, we leverage the additive secret sharing theory to develop secure functions for every layer in an object-detection convolutional neural network (CNN). A vehicle’s sensor data is split and encrypted into multiple secret shares, each of which is processed on an edge server by going through the secure layers of a detection network. The detection results can only be obtained by combining the partial results from the participating edge servers. We have developed proof-of-concept detection networks with secure layers: vePOD Faster R-CNN (two-stage detection) and vePOD YOLO (single-stage detection). Experimental results on public datasets show that vePOD does not degrade the accuracy of object detection and, most importantly, it protects data privacy for vehicles. The execution of a vePOD object-detection network with secure layers is orders of magnitude faster than the existing approaches for data privacy. To the best of our knowledge, this is the first work that targets privacy protection in object-detection tasks with vehicle–edge cooperative computing.