Many problems in multiagent networks can be solved through distributed learning (state estimation) of linear dynamical systems. In this paper, we develop a partial-diffusion Kalman filtering (PDKF) algorithm, as a fully distributed solution for state estimation in the multiagent networks with limited communication resources. In the PDKF algorithm, every agent (node) is allowed to share only a subset of its intermediate estimate vectors with its neighbors at each iteration, reducing the amount of internode communications. We analyze the stability of the PDKF algorithm and show that the algorithm is stable and convergent in both mean and mean-square senses. We also derive a closed-form expression for the steady-state mean-square deviation criterion. Furthermore, we show theoretically and by numerical examples that the PDKF algorithm provides a trade-off between the estimation performance and the communication cost that is extremely profitable.