This paper presents three Moving Horizon Estimation (MHE) methods for discrete-time partitioned linear systems, which are decomposed into coupled subsystems with non-overlapping states. The MHE approach is used because it can exploit physical constraints on states during estimation. Each subsystem solves a reduced-order MHE problem to estimate its own state, and the algorithms differ in terms of communication requirements, accuracy, and computational complexity. The convergence of the estimation error to zero is analyzed in all cases. The three proposed algorithms, PMHE1, PMHE2, and PMHE3, are designed for linear constrained systems decomposed into interconnected subsystems. PMHE1 and PMHE2 decentralize the MHE scheme from [16], while PMHE3 is inspired by the MHE strategy for unconstrained systems from [1]. Decentralization is achieved through approximations of covariance matrices, resulting in suboptimal estimation algorithms compared to centralized MHE. However, convergence conditions are provided for the PMHE schemes. The paper also discusses the design of the PMHE1 algorithm, including offline and online steps for state estimation. The algorithms are applied to partitioned large-scale systems, and sufficient conditions for convergence are established. The three solutions differ in communication requirements, accuracy, and computational complexity. PMHE1 uses a partially connected communication graph, while PMHE2 and PMHE3 assume all-to-all communication but with reduced information transmission. PMHE3 requires no information on noise variances and has constant weights in the cost function, leading to reduced transmission and computational load, though with a loss in noise filtering performance. The paper concludes with a discussion of the results and their implications for distributed state estimation.This paper presents three Moving Horizon Estimation (MHE) methods for discrete-time partitioned linear systems, which are decomposed into coupled subsystems with non-overlapping states. The MHE approach is used because it can exploit physical constraints on states during estimation. Each subsystem solves a reduced-order MHE problem to estimate its own state, and the algorithms differ in terms of communication requirements, accuracy, and computational complexity. The convergence of the estimation error to zero is analyzed in all cases. The three proposed algorithms, PMHE1, PMHE2, and PMHE3, are designed for linear constrained systems decomposed into interconnected subsystems. PMHE1 and PMHE2 decentralize the MHE scheme from [16], while PMHE3 is inspired by the MHE strategy for unconstrained systems from [1]. Decentralization is achieved through approximations of covariance matrices, resulting in suboptimal estimation algorithms compared to centralized MHE. However, convergence conditions are provided for the PMHE schemes. The paper also discusses the design of the PMHE1 algorithm, including offline and online steps for state estimation. The algorithms are applied to partitioned large-scale systems, and sufficient conditions for convergence are established. The three solutions differ in communication requirements, accuracy, and computational complexity. PMHE1 uses a partially connected communication graph, while PMHE2 and PMHE3 assume all-to-all communication but with reduced information transmission. PMHE3 requires no information on noise variances and has constant weights in the cost function, leading to reduced transmission and computational load, though with a loss in noise filtering performance. The paper concludes with a discussion of the results and their implications for distributed state estimation.