The spiking neural network (SNN) is distinguished by its ultra-low power consumption, making it attractive for resource-limited edge intelligence. This paper investigates an energy-efficient (EE) distributed SNN, where multiple edge nodes, each containing a subset of spiking neurons, collaborate to gather and process information through wireless channels. To leverage the benefits of the joint design of neuromorphic computing and wireless communications, we develop quantitative system models and formulate the problem of minimizing the energy consumption of edge devices under constraints of limited bandwidth and spike loss probability. Particularly, a simplified homogeneous SNN is first explored, where the system is proved to have stationary states with a constant firing rate and an alternating optimization based algorithm is proposed for jointly allocating the computation and communication resources. The algorithms are further extended to heterogeneous SNNs by exploiting the statistics of spikes. Extensive simulation results on neuromorphic datasets demonstrate that the developed algorithms can significantly reduce the power consumption of edge systems while ensuring inference accuracy. Moreover, SNNs achieve comparable performance with state-ofthe-art recurrent neural networks (RNNs) but are much more bandwidth-efficient and energy-saving.