The paper deals with decentralized state estimation for spatially distributed systems described by linear partial differential equations from discrete in-space-and-time noisy measurements provided by sensors deployed over the spatial domain of interest. A fully scalable approach is pursued by decomposing the domain into possibly overlapping subdomains assigned to different processing nodes interconnected to form a network. Each node runs a local finite-dimensional discrete-time Kalman filter which exploits the finite element approach for spatial discretization, a backward Euler method for time-discretization and the parallel Schwarz method to iteratively enforce continuity of the field predictions over the boundaries of adjacent subdomains. Numerical stability of the adopted approximation scheme and stability of the proposed distributed finite element Kalman filter are mathematically proved. The effectiveness of the proposed approach is then demonstrated via simulation experiments concerning the estimation of a bi-dimensional temperature field.