This article investigates the optimally distributed consensus control problem for discrete-time multiagent systems with completely unknown dynamics and computational ability differences. The problem can be viewed as solving nonzero-sum games with distributed reinforcement learning (RL), and each agent is a player in these games. First, to guarantee the real-time performance of learning algorithms, a data-based distributed control algorithm is proposed for multiagent systems using offline system interaction data sets. By utilizing the interactive data produced during the run of a real-time system, the proposed algorithm improves system performance based on distributed policy gradient RL. The convergence and stability are guaranteed based on functional analysis and the Lyapunov method. Second, to address asynchronous learning caused by computational ability differences in multiagent systems, the proposed algorithm is extended to an asynchronous version in which executing policy improvement or not of each agent is independent of its neighbors. Furthermore, an actor-critic structure, which contains two neural networks, is developed to implement the proposed algorithm in synchronous and asynchronous cases. Based on the method of weighted residuals, the convergence and optimality of the neural networks are guaranteed by proving the approximation errors converge to zero. Finally, simulations are conducted to show the effectiveness of the proposed algorithm.