This paper studies the problem of distributed state estimation for discrete-time systems over a sensor network through unreliable transmission channels subject to diverse packet dropout probabilities. Each sensor node, equipped with an infinite-time horizon H-infinity Gaussian filter to reach a trade-off between inherently conflicting optimality and robustness, locally constructs an estimate based on its own observation and on those collected from its neighbors over lossy links. A non-zero-sum Nash game is used to deal with such a multiobjective distributed filtering problem. Stabilizing solutions in the mean square (MS) sense are established for a set of cross-coupled modified algebraic Riccati equations associated with each sensor node. Based on the MS stabilizing solutions, causal and bounded Nash equilibrium strategies, consisting of the optimal filter gains and the corresponding worst-case disturbance signals, are further analytically conducted. Moreover, to improve the cohesiveness among different local estimates, an additional consensus objective is taken into consideration by exchanging prior estimates among neighboring nodes. Through a modified algebraic Riccati inequality, an upper bound of the consensus parameter, under which the error dynamics are shown to be MS stable, is derived in terms of the complexity of the stochastic graph associated with the sensor network and the disturbance attenuation level of a stochastic system constructed from the error dynamics. Finally, a numerical example is included to show the validity of the current results.