Resistive switching (RS) of (001) epitaxial multiferroic BiFeO3/La0.67Sr0.33MnO3/SrTiO3 heterostructures is investigated for varying lengths scales in both the thickness and lateral directions. Macroscale current-voltage analyses in conjunction with local conduction atomic force microscopy (CAFM) reveal that whilst both the local and global resistive states are strongly driven by polarization direction, the type of conduction mechanism is different for each distinct thickness regime. Electrode-area dependent studies confirm the RS is dominated by an interface mechanism and not by filamentary formation. Furthermore, CAFM maps allow deconvolution of the roles played by domains and domain walls during the RS process. It is shown that the net polarization direction, and not domain walls, controls the conduction process. An interface mechanism based on barrier height and width alteration due to polarization reversal is proposed, and the role of electronic reconstruction at the interface is further investigated.