This paper addresses the problem of designing distributed state estimation solutions for a network of interconnected systems modelled by linear time-varying (LTV) dynamics in a discrete-time framework. The problem is formulated as a classical optimal estimation problem, for the global system, subject to a given sparsity constraint on the filter gain, which reflects the distributed nature of the network. Two methods are presented, both of them able to compute a sequence of well-performing stabilising gains. Moreover, both methods are validated by resorting to simulations of: (i) a randomly generated synthetic LTV system; and (ii) a large-scale nonlinear network of interconnected tanks. One of the proposed methods relies on a computationally efficient solution, thus it is computed very rapidly. The other achieves better performance, but it is computationally more expensive and requires that a window of the future dynamics of the system is known. When implemented to a nonlinear network, approximated by an LTV system, the proposed methods are able to compute well-performing gains that stabilise the estimation error dynamics. Both algorithms are scalable, being adequate for implementation in large-scale networks.