This paper presents three novel Moving Horizon Estimation (MHE) methods for discrete-time partitioned linear systems, i.e. systems decomposed into coupled subsystems with non-overlapping states. The MHE approach is used due to its capability of exploiting physical constraints on states and noise in the estimation process. In the proposed algorithms, each subsystem solves reduced-order MHE problems to estimate its own state and different estimators have different computational complexity, accuracy and transmission requirements among subsystems. In all cases, proper tuning of the design parameters, i.e. the penalties on the states at the beginning of the estimation horizon, guarantees convergence of the estimation error to zero. Numerical simulations demonstrate the viability of the approach.