Blue straggler star (BSS) candidates have been observed in all old dwarf spheroidal galaxies (dSphs), however whether or not they are authentic BSSs or young stars has been a point of debate. To both address this issue and obtain a better understanding of the formation of BSSs in different environments, we have analysed a sample of BSS candidates in two nearby Galactic dSphs, Draco and Ursa Minor. We have determined their radial and luminosity distributions from wide field multicolour imaging data extending beyond the tidal radii of both galaxies. BSS candidates are uniformly distributed through the host galaxy, whereas a young population is expected to show a more clumpy distribution. Furthermore, the observed radial distribution of BSSs, normalized to both red giant branch (RGB) and horizontal branch (HB) stars, is almost flat, with a slight decrease towards the centre. Such a distribution is at odds with the predictions for a young stellar population, which should be more concentrated. Instead, it is consistent with model predictions for BSS formation by mass transfer in binaries (MT-BSSs). Such results, although not decisive, suggest that these candidates are indeed BSSs and that MT-BSSs form in the same way in Draco and Ursa Minor as in globular clusters. This favours the conclusion that Draco and Ursa Minor are truly `fossil' galaxies, where star formation ceased completely more than 8 billion years ago.