The cloud-based solutions are becoming inefficient due to considerably large time delays, high power consumption, and security and privacy concerns caused by billions of connected wireless devices and typically zillions of bytes of data they produce at the network edge. A blend of edge computing and Artificial Intelligence (AI) techniques could optimally shift the resourceful computation servers closer to the network edge, which provides the support for advanced AI applications (e.g., video/audio surveillance and personal recommendation system) by enabling intelligent decision making on computing at the point of data generation as and when it is needed, and distributed Machine Learning (ML) with its potential to avoid the transmission of the large dataset and possible compromise of privacy that may exist in cloud-based centralized learning. Besides, the deployment of AI techniques to redesign end-to-end communication is attracting attention to improve communication performance. Therefore, the interaction of AI and wireless communications generates a new concept, named native AI wireless networks. In this paper, we conduct a comprehensive overview of recent advances in distributed intelligence in wireless networks under the umbrella of native AI wireless networks, with a focus on the design of distributed learning architectures for heterogeneous networks, on AI-enabled edge computing, on the communication-efficient technologies to support distributed learning, and on the AI-empowered end-to-end communications. We highlight the advantages of hybrid distributed learning architectures compared to state-of-the-art distributed learning techniques. We summarize the challenges of existing research contributions in distributed intelligence in wireless networks and identify potential future opportunities.