This paper presents a distributed filtering algorithm for networked agents modeled with independent linear state models and measurements. In addition to local state measurements, agents utilize relative state measurements and communicated state estimates to improve their individual state estimates. These kind of measurement models are motivated by the problem of localization in sensor networks and multi-vehicle robotics. In this paper, we explicitly consider that all the state models, measurements and communications are subject to disturbances and we propose a minimum-energy filtering approach to derive the state estimates. The filters are distributed due to the practical requirement that the agents are only allowed to exchange relative measurements and communications in their local neighbourhoods. Sufficient conditions guaranteeing the convergence of the proposed distributed algorithms are provided in terms of detectability conditions and also in terms of bounds on information sharing among the agents. A numerical example is provided that illustrates the performance of the proposed algorithm and also reinforces the importance of the convergence conditions.