To fully utilize the spatial multiplexing gains or array gains of massive MIMO, the channel state information must be obtained at the transmitter side (CSIT). However, conventional CSIT estimation approaches are not suitable for FDD massive MIMO systems because of the overwhelming training and feedback overhead. In this paper, we consider multi-user massive MIMO systems and deploy the compressive sensing (CS) technique to reduce the training as well as the feedback overhead in the CSIT estimation. The multi-user massive MIMO systems exhibits a hidden joint sparsity structure in the user channel matrices due to the shared local scatterers in the physical propagation environment. As such, instead of naively applying the conventional CS to the CSIT estimation, we propose a distributed compressive CSIT estimation scheme so that the compressed measurements are observed at the users locally, while the CSIT recovery is performed at the base station jointly. A joint orthogonal matching pursuit recovery algorithm is proposed to perform the CSIT recovery, with the capability of exploiting the hidden joint sparsity in the user channel matrices. We analyze the obtained CSIT quality in terms of the normalized mean absolute error, and through the closed-form expressions, we obtain simple insights into how the joint channel sparsity can be exploited to improve the CSIT recovery performance.