The deep autoencoder (DAE) framework has turned out to be efficient in reducing the channel state information (CSI) feedback overhead in massive multiple-input multipleoutput (mMIMO) systems. However, these DAE approaches presented in prior works rely heavily on large-scale data collected through the base station (BS) for model training, thus rendering excessive bandwidth usage and data privacy issues, particularly for mMIMO systems. When considering users' mobility and encountering new channel environments, the existing CSI feedback models may often need to be retrained. Returning back to previous environments, however, will make these models perform poorly and face the risk of catastrophic forgetting. To solve the above challenging problems, we propose a novel gossiping generative adversarial network (Gossip-GAN)-aided CSI feedback training framework. Notably, Gossip-GAN enables the CSI feedback training with low-overhead while preserving users' privacy. Specially, each user collects a small amount of data to train a GAN model. Meanwhile, a fully distributed gossip-learning strategy is exploited to avoid model overfitting, and to accelerate the model training as well. Simulation results demonstrate that Gossip-GAN can i) achieve a similar CSI feedback accuracy as centralized training with real-world datasets, ii) address catastrophic forgetting challenges in mobile scenarios, and iii) greatly reduce the uplink bandwidth usage. Besides, our results show that the proposed approach possesses an inherent robustness.