We examine the star formation properties of bright (~0.1 L*) satellites around isolated ~L* hosts in the local Universe using spectroscopically confirmed systems in the Sloan Digital Sky Survey Data Release 7. Our selection method is carefully designed with the aid of Nbody simulations to avoid groups and clusters. We find that satellites are significantly more likely to be quenched than a stellarmass-matched sample of isolated galaxies. Remarkably, this quenching occurs only for satellites of hosts that are themselves quenched: while star formation is unaffected in the satellites of star-forming hosts, satellites around quiescent hosts are more than twice as likely to be quenched than stellar-mass-matched field samples. One implication of this is that whatever shuts down star formation in isolated, passive L* galaxies also play at least an indirect role in quenching star formation in their bright satellites. The previously reported tendency for 'galactic conformity' in colour/morphology may be a by-product of this host-specific quenching dichotomy. The Sérsic indices of quenched satellites are statistically identical to those of field galaxies with the same specific star formation rates, suggesting that environmental and secular quenching give rise to the same morphological structure. By studying the distribution of pairwise velocities between the hosts and satellites, we find dynamical evidence that passive host galaxies reside in dark matter haloes that are~45 per cent more massive than those of star-forming host galaxies of the same stellar mass.We emphasize that even around passive hosts, the mere fact that galaxies become satellites does not typically result in star formation quenching: we find that only ~30 per cent of ~0.1L* galaxies that fall in from the field are quenched around passive hosts, compared with ~0 per cent around star-forming hosts. © 2013 The Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society.