Edge-cloud collaborative inference empowers resource-limited IoT devices to support deep learning applications without disclosing their raw data to the cloud server, thus preserving privacy. Nevertheless, prior research has shown that collaborative inference still results in the exposure of data and predictions from edge devices. To enhance the privacy of collaborative inference, we introduce a defense strategy called PrivaScissors, which is designed to reduce the mutual information between a model's intermediate outcomes and the device's data and predictions. We evaluate PrivaScissors's performance on several datasets in the context of diverse attacks and offer a theoretical robustness guarantee.