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Using the diffusion strategies, an unknown parameter vector can be estimated over an adaptive network by combining the intermediate estimates of neighboring nodes at each node. We propose an extension to the diffusion recursive least-squares algorithm by allowing partial sharing of the entries of the intermediate estimate vectors among the neighbors. Accordingly, the proposed algorithm, termed partial-diffusion recursive least-squares (PDRLS), enables a trade-off between estimation performance and communication cost. We analyze the performance of the PDRLS algorithm and prove its convergence in both mean and mean-square senses. We also derive a theoretical expression for its steady-state mean-square deviation. Simulation results substantiate the efficacy of the PDRLS algorithm and demonstrate a good match between theory and experiment.