Context. Planet perturbations have been often invoked as a potential explanation for many spatial structures that have been imaged in debris discs. So far this issue has been mostly investigated with pure N-body numerical models, which neglect the crucial effect collisions within the disc can have on the disc's response to dynamical perturbations. Aims. We numerically investigate how the coupled effect of collisions and radiation pressure can affect the formation and survival of radial and azimutal structures in a disc perturbed by a planet. We consider two different set-ups: a planet embedded within an extended disc and a planet exterior to an inner debris ring. One important issue we want to address is under which conditions a planet's signature can be observable in a collisionally active disc. Methods. We use our DyCoSS code, which is designed to investigate the structure of perturbed debris discs at dynamical and collisional steady-state, and derive synthetic images of the system in scattered light. The planet's mass and orbit, as well as the disc's collisional activity (parameterized by its average vertical optical depth τ 0 ) are explored as free parameters. Results. We find that collisions always significantly damp planet-induced spatial structures. For the case of an embedded planet, the planet's signature, mostly a density gap around its radial position, should remain detectable in head-on images if M planet ≥ M Saturn . If the system is seen edge-on, however, inferring the presence of the planet is much more difficult, as only weak asymmetries remain in a collisionally active disc, although some planet-induced signatures might be observable under very favourable conditions. For the case of an inner ring and an external planet, planetary perturbations cannot prevent collision-produced small fragments from populating the regions beyond the ring. The radial luminosity profile exterior to the ring is in most cases close to the one it should have in the absence of the external planet. The most significant signature left by a Jovian planet on a circular orbit are precessing azimutal structures that can be used to indirectly infer its presence. For a planet on an eccentric orbit, we show that the ring becomes elliptic and that the well known pericentre glow effect is visible despite of collisions and radiation pressure, but that detecting such features in observed discs is not an unambiguous indicator of the presence of an outer planet.