On-device intelligence (ODI) enables artificial intelligence (AI) applications to run on end devices, providing real-time and customized AI inference without relying on remote servers. However, training models for on-device deployment face significant challenges due to the decentralized and privacy-sensitive nature of users' data, along with end-side constraints related to network connectivity, computation efficiency, etc. Existing training paradigms, such as cloud-based training, federated learning, and transfer learning, fail to sufficiently address these practical constraints that are prevalent for devices. To overcome these challenges, we propose Privacy-Enhanced Training-as-a-Service (PTaaS), a novel service computing paradigm that provides privacy-friendly, customized AI model training for end devices. PTaaS outsources the core training process to remote and powerful cloud or edge servers, efficiently developing customized on-device models based on uploaded anonymous queries, enhancing data privacy while reducing the computation load on individual devices. We explore the definition, goals, and design principles of PTaaS, alongside emerging technologies that support the PTaaS paradigm. An architectural scheme for PTaaS is also presented, followed by a series of open problems that set the stage for future research directions in the field of PTaaS.