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A recent emphasis of distributed learning research has been on federated learning (FL), in which model training is conducted by the data-collecting devices. Existing research on FL has mostly focused on a star topology learning architecture with synchronized (time-triggered) model training rounds, where the local models of the devices are periodically aggregated by a centralized coordinating node. However, in many settings, such a coordinating node may not exist, motivating efforts to fully decentralize FL. In this work, we propose a novel methodology for distributed model aggregations via asynchronous, event-triggered consensus iterations over the network graph topology. We consider heterogeneous communication event thresholds at each device that weigh the change in local model parameters against the available local resources in deciding the benefit of aggregations at each iteration. Through theoretical analysis, we demonstrate that our methodology achieves asymptotic convergence to the globally optimal learning model under standard assumptions in distributed learning and graph consensus literature, and without restrictive connectivity requirements on the underlying topology. Subsequent numerical results demonstrate that our methodology obtains substantial improvements in communication requirements compared with FL baselines.