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Machine Learning as a Service (MLaaS) operators provide model training and prediction on the cloud. MLaaS applications often rely on centralised collection and aggregation of user data, which could lead to significant privacy concerns when dealing with sensitive personal data. To address this problem, we propose PrivEdge, a technique for privacy-preserving MLaaS that safeguards the privacy of users who provide their data for training, as well as users who use the prediction service. With PrivEdge, each user independently uses their private data to locally train a one-class reconstructive adversarial network that succinctly represents their training data. As sending the model parameters to the service provider in the clear would reveal private information, PrivEdge secret-shares the parameters among two non-colluding MLaaS providers, to then provide cryptographically private prediction services through secure multi-party computation techniques. We quantify the benefits of PrivEdge and compare its performance with state-of-the-art centralised architectures on three privacy-sensitive image-based tasks: individual identification, writer identification, and handwritten letter recognition. Experimental results show that PrivEdge has high precision and recall in preserving privacy, as well as in distinguishing between private and non-private images. Moreover, we show the robustness of PrivEdge to image compression and biased training data. The source code is available at https://github.com/smartcameras/PrivEdge.