Downlink joint transmission by a cluster of remote radio heads (RRHs) is an essential technique for enhancing throughput in future cellular networks. This method requires global channel state information (CSI) at the processing unit that designs the joint precoder. To this end, a large amount of CSI must be shared between the RRHs and that unit. This paper proposes two contributions. The first is a new upper bound on the rate loss, which implies a lower bound on the achievable rate, obtained by a cluster of RRHs that employ joint zero-forcing (ZF) with incomplete CSI. The second contribution, which follows insights from the bound, is a new CSI sharing scheme that drastically reduces the large overhead associated with acquiring global CSI for joint transmission. In a nutshell, each RRH applies a local precoding matrix that creates low-dimensional effective channels that can be quantized more accurately with fewer bits, thereby reducing the overhead of sharing CSI. In addition to the CSI sharing-overhead, this scheme reduces the data rate that must be delivered to each RRH in the cluster.