We present a novel distributed Kalman-like observer for cooperative state estimation in multi-agent systems. Our approach builds on a class of existing Kalmanlike observers that replace the process covariance matrix with a forgetting factor. We show that this replacement enables the propagation of the information matrix dynamics in a fully distributed manner, while preserving key stability properties. We compute the observers correction term by solving a linear equation dynamically in a distributed manner, circumventing the need for direct centralized matrix inversion. Unlike existing methods that partially discard cross-information to allow distributed computations, our approach preserves inter-agent coupling. Rigorous stability guarantees are provided, and numerical simulations in a cooperative localization scenario demonstrate the effectiveness of the approach in estimating agent states.