In this paper, we propose a stochastic model predictive control (MPC) algorithm for linear distributed discrete-time systems affected by unbounded additive Gaussian disturbances, which are subject to local probabilistic constraints. Probabilistic constraints are treated with the concept of probabilistic reachable sets, which are an analogy to robust reachable sets for robust MPC. We present a method which decouples the global covariance matrix into a block diagonal upper bound. Together with the decomposition of the centralized problem, we define local probabilistic invariant sets as terminal regions, where we additionally derive a condition that gives us a probabilistic guarantee of invariance. We demonstrate our approach on an example, highlighting the closed-loop performance and constraint satisfaction compared to a centralized scheme.