Two electrons in planar Penning traps can be coupled by connecting the trap electrodes with a metallic wire. The wire provides a capacitive and resistive link between the image charges that the electron oscillation (axial motion) is inducing in the electrodes. This can be operated as a quantum network to transfer excitation quanta between the traps and to entangle the two electrons. We give a detailed analysis of this system in the dispersive limit and assess the impact of resistive thermal noise. The latter is found to be remarkably small at temperatures below 4 K in miniaturized planar traps (sizes below 1 mm). The master equation for the two electron system in the presence of resistive noise is solved exactly. We compute fidelities for basic quantum gates needed in quantum information processing.