Federated Learning (FL) enables end-user devices to collaboratively train ML models without sharing raw data, thereby preserving data privacy. In FL, a central parameter server coordinates the learning process by iteratively aggregating the trained models received from clients. Yet, deploying a central server is not always feasible due to hardware unavailability, infrastructure constraints, or operational costs. We present Plexus, a fully decentralized FL system for large networks that operates without the drawbacks originating from having a central server. Plexus distributes the responsibilities of model aggregation and sampling among participating nodes while avoiding network-wide coordination. We evaluate Plexus using realistic traces for compute speed, pairwise latency and network capacity. Our experiments on three common learning tasks and with up to 1000 nodes empirically show that Plexus reduces time-to-accuracy by 1.4-1.6×, communication volume by 15.8-292× and training resources needed for convergence by 30.5-77.9× compared to conventional decentralized learning algorithms.