We present an information-theoretic perspective to group fairness trade-offs in federated learning (FL) with respect to sensitive attributes, such as gender, race, etc. Existing works mostly focus on either \emph{global fairness} (overall disparity of the model across all clients) or \emph{local fairness} (disparity of the model at each individual client), without necessarily considering their trade-offs. There is a lack of understanding of the interplay between global and local fairness in FL, and if and when one implies the other. To address this gap, we leverage a body of work in information theory called partial information decomposition (PID) which first identifies three sources of unfairness in FL, namely, \emph{Unique Disparity}, \emph{Redundant Disparity}, and \emph{Masked Disparity}. Using canonical examples, we demonstrate how these three disparities contribute to global and local fairness. This decomposition helps us derive fundamental limits and trade-offs between global or local fairness, particularly under data heterogeneity, as well as, derive conditions under which one implies the other. We also present experimental results on real-world datasets to support our theoretical findings. This work offers a more nuanced understanding of the sources of disparity in FL that can inform the use of local disparity mitigation techniques, and their convergence and effectiveness when deployed in practice.