Privacy laws like GDPR necessitate effective approaches to safeguard data privacy. Existing works on data privacy protection of DNNs mainly concentrated on the model training phase. However, these approaches become impractical when dealing with the outsourcing of sensitive data. Furthermore, they have encountered significant challenges in balancing the utility-privacy trade-off. How can we generate privacy-preserving surrogate data suitable for usage or sharing without a substantial performance loss? In this paper, we concentrate on a realistic scenario, where sensitive data must be entrusted to a third party for the development of a deep learning service. We introduce a straightforward yet highly effective framework for the practical protection of privacy in visual data via veiled examples. Our underlying concept involves segregating the privacy information present in images into two distinct categories: the privacy information perceptible at the human visual level (i.e., Human-perceptible Info) and the latent features recognized by DNN models during training and prediction (i.e., DNN-perceptible Info). Compared with the original data, the veiled data conserves the latent features while obfuscating the visual privacy information. Just like a learnable veil that is usable for DNNs but invisible for humans, the veiled data can be used for training and prediction of models. More importantly, models trained with the veiled data can effectively recognize the original data. Extensive evaluations of various datasets and models show the effectiveness and security of the Veil Privacy framework.