To achieve a high learning accuracy, generative adversarial networks (GANs) must be fed by large datasets that adequately represent the data space. However, in many scenarios, the available datasets may be limited and distributed across multiple agents, each of which is seeking to learn the distribution of the data on its own. In such scenarios, the agents often do not wish to share their local data as it can cause communication overhead for large datasets. In this paper, to address this multi-agent GAN problem, a novel brainstorming GAN (BGAN) architecture is proposed using which multiple agents can generate real-like data samples while operating in a fully distributed manner. BGAN allows the agents to gain information from other agents without sharing their real datasets but by ``brainstorming'' via the sharing of their generated data samples. In contrast to existing distributed GAN solutions, the proposed BGAN architecture is designed to be fully distributed, and it does not need any centralized controller. Moreover, BGANs are shown to be scalable and not dependent on the hyperparameters of the agents' deep neural networks (DNNs) thus enabling the agents to have different DNN architectures. Theoretically, the interactions between BGAN agents are analyzed as a game whose unique Nash equilibrium is derived. Experimental results show that BGAN can generate real-like data samples with higher quality and lower Jensen-Shannon divergence (JSD) and Frèchet Inception distance (FID) compared to other distributed GAN architectures.