This paper proposes a distributed method for jointly estimating the input and the state of a given system observed through a sensor network. Traditionally, an unbiased state estimation can be obtained by using distributed Kalman filters if the system is subject to noise with known stochastic properties. Unfortunately, if the system is subject to a completely unknown input, representing faults, unknown disturbances or unmodeled dynamics, the estimated state is no longer unbiased. This study proposes new distributed filters that allow carrying out an unbiased state estimation in the presence of the unknown input. In addition, the proposed filters provide an estimation of the unknown inputs. Herein, it is assumed that the unknown input affects sensor measurements as well as system states. Two consensus-based distributed estimation algorithms are provided in this paper. The first algorithm gives an optimal minimum variance estimation if perfect consensus is achieved while the second algorithm provides a suboptimal solution with less computation effort.