To keep pace with the explosive growth of Artificial Intelligence (AI) and Machine Learning (ML)-dominated applications, distributed intelligence solutions are gaining momentum, which exploit cloud facilities, edge nodes and end-devices to increase the overall computational power, meet application requirements, and optimize performance. Despite the benefits in terms of data privacy and efficient usage of resources, distributing intelligence throughout the cloud-to-things continuum raises unprecedented challenges to the network design. Distributed AI/ML components need high-bandwidth, low-latency connectivity to execute learning and inference tasks, while ensuring high-accuracy and energy-efficiency. This paper aims to explore the new challenging distributed intelligence scenario by extensively and critically scanning the main research achievements in the literature. In addition, starting from them, the main building blocks of a network ecosystem that can enable distributed intelligence are identified and the authors' views are dissected to provide guidelines for the design of a ''future network for distributed Intelligence''.