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Reconfigurable Intelligent Surfaces (RISs) are recently gaining remarkable attention as a low-cost, hardware-efficient, and highly scalable technology capable of offering dynamic control of electro-magnetic wave propagation. Their envisioned dense deployment over various obstacles of the, otherwise passive, wireless communication environment has been considered as a revolutionary means to transform them into network entities with reconfigurable properties, providing increased environmental intelligence for diverse communication objectives. One of the major challenges with RIS-empowered wireless communications is the low-overhead dynamic configuration of multiple RISs, which according to the current hardware designs have very limited computing and storage capabilities. In this paper, we consider a typical communication pair between two nodes that is assisted by a plurality of RISs, and devise low-complexity supervised learning approaches for the RISs' phase configurations. By assuming common tunable phases in groups of each RIS's unit elements, we present multi-layer perceptron Neural Network (NN) architectures that can be trained either with positioning values or the instantaneous channel coefficients. We investigate centralized and individual training of the RISs, as well as their federation, and assess their computational requirements. Our simulation results, including comparisons with the optimal phase configuration scheme, showcase the benefits of adopting individual NNs at RISs for the link budget performance boosting.