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Low-latency data processing is essential for wide-area monitoring of smart grids. Distributed and local data processing is a promising approach for enabling low-latency requirements and avoiding the large overhead of transferring large volumes of time-sensitive data to central processing units. State estimation in power systems is one of the key functions in wide-area monitoring, which can greatly benefit from distributed data processing and improve real-time system monitoring. In this paper, data-driven Kalman filters have been used for multi-area distributed state estimation. The presented state estimation approaches are data-driven and model-independent. The design phase is offline and involves modeling multivariate time-series measurements from PMUs using linear and non-linear system identification techniques. The measurements of the phase angle, voltage, reactive and real power are used for next-step prediction of the state of the buses. The performance of the presented data-driven, distributed state estimation techniques are evaluated for various numbers of regions and modes of information sharing on the IEEE 118 test case system.