Magnetic field perturbations measured during Galileo flybys of Europa and Callisto are consistent with dipole fields induced by the temporal variations of the ambient jovian magnetospheric field. These fields are close to those expected for perfectly conducting moons. We investigate the implications of these observations for the electrical structure of the moon's interiors using a simple shell model. It is found that Europa and Callisto must possess regions where the conductivity exceeds 0.06 and 0.02 S/m at a depth of less than 200 and 300 km below the surface, respectively. This conductivity is unattainable in ice or silicates, unless the ice layer is at least partially molten or very large temperature gradients can be maintained below the ice. An ionosphere or a cloud of pick-up ions are probably also insufficiently conductive. Global Earth-like oceans under the surface of both moons could account for the observations provided they are at least a few kilometers thick.