We consider the problem of fixed-point distributed Kalman smoothing, where a set of nodes are required to estimate the initial condition of a certain process based on their measurements of the evolution of the process. Specifically, we consider linear state-space models where the Kalman smoother gives us the MMSE estimate of the initial state of the system. We propose distributed diffusion solutions where nodes communicate with their neighbors and information is propagated through the network via a diffusion process. Hierarchical cooperation schemes are also described.