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We compute the electrical current flow and associated Oersted fields in magnetic nanocontact (NC) systems with finite-element simulations. Underneath the nanocontact, the current distribution is found to be mainly perpendicular to the film plane, while the flow is essentially in the film plane outside the nanocontact region. For circular nanocontacts, we provide criteria for which the cylindrical approximation remains valid for electrical flow in asymmetric rectangular devices. The nature of the current distribution, and the induced field, is a key to understanding spin-torque driven magnetization dynamics in such systems.