In the framework of efficient partitioned numerical schemes for simulating multiphysics PDE problems, we propose using intergrid transfer operators based on radial basis functions to accurately exchange information among different PDEs defined in the same computational domain. Different (potentially non-nested) meshes can be used for the space discretization of the PDEs. The projection of the (primary) variables that are shared by the different PDEs (through the coupling terms) is carried out with Rescaled Localized Radial Basis Functions. We validate our approach by a numerical test for which we also show the scalability of the intergrid transfer operator in the framework of high performance computing. Then, we apply it to the electromechanical model for the human heart function, and simulate a heartbeat of an idealized left ventricle. We show that our approach enables the solution of large-scale multiphysics problems, especially when the individual models exhibit very different spatial scales.