Balancing the needs of data privacy and predictive utility is a central challenge for machine learning in healthcare. In particular, privacy concerns have led to a dearth of public datasets, complicated the construction of multi-hospital cohorts and limited the utilization of external machine learning resources. To remedy this, new methods are required to enable data owners, such as hospitals, to share their datasets publicly, while preserving both patient privacy and modeling utility. We propose NeuraCrypt, a private encoding scheme based on random deep neural networks. NeuraCrypt encodes raw patient data using a randomly constructed neural network known only to the data-owner, and publishes both the encoded data and associated labels publicly. From a theoretical perspective, we demonstrate that sampling from a sufficiently rich family of encoding functions offers a well-defined and meaningful notion of privacy against a computationally unbounded adversary with full knowledge of the underlying data-distribution. We propose to approximate this family of encoding functions through random deep neural networks. Empirically, we demonstrate the robustness of our encoding to a suite of adversarial attacks and show that NeuraCrypt achieves competitive accuracy to non-private baselines on a variety of x-ray tasks. Moreover, we demonstrate that multiple hospitals, using independent private encoders, can collaborate to train improved x-ray models. Finally, we release a challenge dataset to encourage the development of new attacks on NeuraCrypt.