Deriving efficient cooperative policies in uncertain dynamic environments poses huge challenges for a distributed swarm system due to the limited capability of the agents and the complex dynamics of the environment. In this paper, a novel distributed method based on deep reinforcement learning using observation-level and communication-level graph networks is proposed to learn cooperative policies for the distributed swarm system. Specifically, a relational directed graph attention neural network is designed to model observation-level graphs composed of heterogeneous relational graphs among each agent and each type of entities (e.g., obstacles, other teammates, opponents), for extracting different relational representations. Moreover, a relevant directed graph attention network is presented to cut off the ineffective communication among irrelevant agents, and model a relevant communication topology between each agent and relevant homogeneous neighbor agents as an communication-level graph, for promoting efficient inter-agent interactions. Furthermore, a distributed actor-critic algorithm with full parameter sharing is implemented to learn cooperative swarm policies by using distributed critics, which avoids the curse of dimensionality under a centralized critic. Various simulation results validate the effectiveness and generalization of the proposed method, and demonstrate that the proposed method outperforms existing state-of-the-art methods on coverage and pursuit tasks.