In this paper, we propose an age of incorrect information (AoII)-aware data dissemination scheme for distributed multi-agent systems (MASs). In the proposed scheme, AoII is utilized to measure the importance of data in terms of timeliness and content. We formulate the joint optimization of time slot allocation and agent selection as a decentralized partially observable Markov decision process (Dec-POMDP), with the objective of minimizing the AoII. To solve the Dec-POMDP, a novel multi-agent reinforcement learning algorithm (DV-MAPPO) is proposed. In particular, to tackle challenges posed by the partial observability of global system information, each agent estimates the global system state using variational inference. Moreover, to improve the accuracy of global system state estimation, each agent is given an intrinsic reward that is dominated by the accuracy of estimates. The proposed data dissemination scheme is implemented and evaluated in various missions. Simulation results show that the proposed data dissemination scheme outperforms traditional data dissemination schemes in terms of AoII. Furthermore, in typical multi-agent collaborative tasks, the proposed scheme facilitates more efficient cooperation among multiple agents compared to the data distribution mechanisms that ignore the importance of data.