We have observed two kinematically offset active galactic nuclei (AGN), whose ionised gas is at a different line-of-sight velocity to their host galaxies, with the SAMI integral field spectrograph (IFS). One of the galaxies shows gas kinematics very different to the stellar kinematics, indicating a recent merger or accretion event. We demonstrate that the star formation associated with this event was triggered within the last 100 Myr. The other galaxy shows simple disc rotation in both gas and stellar kinematics, aligned with each other, but in the central region has signatures of an outflow driven by the AGN. Other than the outflow, neither galaxy shows any discontinuity in the ionised gas kinematics at the galaxy's centre. We conclude that in these two cases there is no direct evidence of the AGN being in a supermassive black hole binary system. Our study demonstrates that selecting kinematically offset AGN from single-fibre spectroscopy provides, by definition, samples of kinematically peculiar objects, but IFS or other data are required to determine their true nature.