Internet-of-Things (IoT) devices and applications are being deployed in our homes and workplaces. These devices often rely on continuous data collection to feed machine learning models. However, this approach introduces several privacy and efficiency challenges, as the service operator can perform unwanted inferences on the available data. Recently, advances in edge processing have paved the way for more efficient, and private, data processing at the source for simple tasks and lighter models, though they remain a challenge for larger and more complicated models. In this article, we present a hybrid approach for breaking down large, complex deep neural networks for cooperative, and privacy-preserving analytics. To this end, instead of performing the whole operation on the cloud, we let an IoT device to run the initial layers of the neural network, and then send the output to the cloud to feed the remaining layers and produce the final result. In order to ensure that the user's device contains no extra information except what is necessary for the main task and preventing any secondary inference on the data, we introduce Siamese fine-tuning. We evaluate the privacy benefits of this approach based on the information exposed to the cloud service. We also assess the local inference cost of different layers on a modern handset. Our evaluations show that by using Siamese fine-tuning and at a small processing cost, we can greatly reduce the level of unnecessary, potentially sensitive information in the personal data, thus achieving the desired tradeoff between utility, privacy, and performance.