When a star gets too close to a supermassive black hole, it is destroyed by the strong tidal forces of the compact object in a tidal disruption event (TDE). As the debris falls back to the disruption site, they undergo complex interactions during which shocks lead to the formation of an accretion disc. This phase and the subsequent gas accretion onto the black hole cause a bright flare of months to years. The rate of TDEs can vary by orders of magnitude depending on the environment and the mechanism that launches the star towards the black hole. For the largest rates, two disruptions can take place shortly one after the other in a double TDE. In this case, the two debris streams may collide with each other resulting in an electromagnetic emission that is absent from single TDEs. We analytically evaluate the conditions for this streams collision to occur. It requires that the difference in pericenter location between the two disruptions makes up for the time delay between them. In addition, the width of the streams must compensate for the vertical offset induced by the inclination of their orbital planes. If the double TDE happens following the tidal separation of a binary, we demonstrate that the streams have a high probability of colliding. For this mechanism, we determine through hydrodynamical simulations that the associated shocks trigger a burst of radiation lasting a few days with a luminosity L ≈ 1043 erg/s. This signal could be detected as a precursor to the main TDE flare and used to get a better handle on the efficiency of accretion disc formation from observations.