The cloud-based solutions are becoming inefficient due to considerably large time delays, high power consumption, security and privacy concerns caused by billions of connected wireless devices and typically zillions 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 large dataset and possible compromise of privacy that may exist in cloud-based centralized learning. Therefore, AI is envisioned to become native and ubiquitous in future communication and networking systems. 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 basic concepts of native-AI wireless networks, on the AI-enabled edge computing, on the design of distributed learning architectures for heterogeneous networks, 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 the state-of-art distributed learning techniques. We summarize the challenges of existing research contributions in distributed intelligence in wireless networks and identify the potential future opportunities.