Currently, numerical optimization methods are used to solve distributed optimal power allocation (OPA) problems for islanded microgrid (MG) systems. Most of them are developed based on rigorous mathematical derivation. However, the complexity of such optimization algorithms inevitably creates a gap between theoretical analysis and real-time implementation. In order to bridge such a gap, in this article we provide a new distributed learning-based framework to solve the real-time OPA problem. Specifically, inspired by the human-thinking scheme, distributed deep neural networks (DNNs) together with a dynamic average consensus algorithm are first employed to obtain an approximate OPA solution in a distributed manner. Then a distributed balance generation and demand algorithm is designed to fine-tune it to obtain the final optimal feasible solution. In addition, it is theoretically proved that the proposed DNN can well approximate one existing OPA algorithm (Guo et al. 2018), where quantitative numbers of at most how many hidden layers and neurons are provided. Several experimental case studies show that our proposed distributed learning framework can achieve similar optimal results to those obtained by using typical existing distributed numerical optimization methods while it is superior in terms of simplicity and real-time capability.