This report presents three Moving Horizon Estimation (MHE) methods for discrete-time partitioned linear systems, where the system is decomposed into coupled subsystems with non-overlapping states. The MHE approach is chosen due to its ability to exploit physical constraints on states during the estimation process. Each subsystem solves a reduced-order MHE problem to estimate its own state, with different estimators having varying computational complexity, accuracy, and communication requirements among subsystems. The conditions for the convergence of the estimation error to zero are analyzed for all proposed methods. The first method, PMHE1, relies on a partially connected communication graph, where links are present only if subsystem dynamics are coupled. The second method, PMHE2, assumes all-to-all communication but transmits less information over each channel. The third method, PMHE3, also assumes all-to-all communication but requires no information on noise variances, leading to reduced transmission and computational load at the cost of a loss in noise filtering performance. The report includes detailed formulations of the optimization problems, convergence properties, and algorithmic summaries for each method. It also discusses the choice of weights and provides sufficient conditions for convergence, including a measure of the quality of the partition. The results extend the convergence analysis of centralized estimators to distributed estimation in the presence of constraints.This report presents three Moving Horizon Estimation (MHE) methods for discrete-time partitioned linear systems, where the system is decomposed into coupled subsystems with non-overlapping states. The MHE approach is chosen due to its ability to exploit physical constraints on states during the estimation process. Each subsystem solves a reduced-order MHE problem to estimate its own state, with different estimators having varying computational complexity, accuracy, and communication requirements among subsystems. The conditions for the convergence of the estimation error to zero are analyzed for all proposed methods. The first method, PMHE1, relies on a partially connected communication graph, where links are present only if subsystem dynamics are coupled. The second method, PMHE2, assumes all-to-all communication but transmits less information over each channel. The third method, PMHE3, also assumes all-to-all communication but requires no information on noise variances, leading to reduced transmission and computational load at the cost of a loss in noise filtering performance. The report includes detailed formulations of the optimization problems, convergence properties, and algorithmic summaries for each method. It also discusses the choice of weights and provides sufficient conditions for convergence, including a measure of the quality of the partition. The results extend the convergence analysis of centralized estimators to distributed estimation in the presence of constraints.