Belief propagation is an algorithm that is known from statistical physics and computer science. It provides an efficient way of calculating marginals that involve large sums of products which are efficiently rearranged into nested products of sums to approximate the marginals. It allows a reliable estimation of the state and its variance of power grids that is needed for the control and forecast of power grid management. At prototypical examples of IEEE grids we show that belief propagation not only scales linearly with the grid size for the state estimation itself but also facilitates and accelerates the retrieval of missing data and allows an optimized positioning of measurement units. Based on belief propagation, we give a criterion for how to assess whether other algorithms, using only local information, are adequate for state estimation for a given grid. We also demonstrate how belief propagation can be utilized for coarse-graining power grids toward representations that reduce the computational effort when the coarse-grained version is integrated into a larger grid. It provides a criterion for partitioning power grids into areas in order to minimize the error of flow estimates between different areas.