Our Galactic Centre, Sagittarius A*, is believed to harbour a supermassive black hole, as suggested by observations tracking individual orbiting stars. Upcoming submillimetre very-long baseline interferometry images of Sagittarius A* carried out by the Event Horizon Telescope collaboration (EHTC) are expected to provide critical evidence for the existence of this supermassive black hole. We assess our present ability to use EHTC images to determine whether they correspond to a Kerr black hole as predicted by Einstein’s theory of general relativity or to a black hole in alternative theories of gravity. To this end, we perform general-relativistic magnetohydrodynamical simulations and use general-relativistic radiative-transfer calculations to generate synthetic shadow images of a magnetized accretion flow onto a Kerr black hole. In addition, we perform these simulations and calculations for a dilaton black hole, which we take as a representative solution of an alternative theory of gravity. Adopting the very-long baseline interferometry configuration from the 2017 EHTC campaign, we find that it could be extremely difficult to distinguish between black holes from different theories of gravity, thus highlighting that great caution is needed when interpreting black hole images as tests of general relativity.