Training a proprietary Neural Network (NN) model with a proprietary dataset on the cloud comes at the risk of exposing the model architecture and the dataset to the cloud service provider. To tackle this problem, in this paper, we present an NN obfuscation framework, called Amalgam, to train NN models in a privacy-preserving manner in existing cloud-based environments. Amalgam achieves that by augmenting NN models and the datasets to be used for training with well-calibrated noise to "hide" both the original model architectures and training datasets from the cloud. After training, Amalgam extracts the original models from the augmented models and returns them to users. Our evaluation results with different computer vision and natural language processing models and datasets demonstrate that Amalgam: (i) introduces modest overheads into the training process without impacting its correctness, and (ii) does not affect the model's accuracy.