The high communication cost of sending model updates from the clients to the server is a significant bottleneck for scalable federated learning (FL). Among existing approaches, state-of-the-art bitrate-accuracy tradeoffs have been achieved using stochastic compression methods -- in which the client $n$ sends a sample from a client-only probability distribution $q_{φ^{(n)}}$, and the server estimates the mean of the clients' distributions using these samples. However, such methods do not take full advantage of the FL setup where the server, throughout the training process, has side information in the form of a global distribution $p_θ$ that is close to the clients' distribution $q_{φ^{(n)}}$ in Kullback-Leibler (KL) divergence. In this work, we exploit this closeness between the clients' distributions $q_{φ^{(n)}}$'s and the side information $p_θ$ at the server, and propose a framework that requires approximately $D_{KL}(q_{φ^{(n)}}|| p_θ)$ bits of communication. We show that our method can be integrated into many existing stochastic compression frameworks to attain the same (and often higher) test accuracy with up to $82$ times smaller bitrate than the prior work -- corresponding to 2,650 times overall compression.