Secret interactions among sterile neutrinos have been recently proposed as an escape route to reconcile eV sterile neutrino hints from short-baseline anomalies with cosmological observations. In particular models with coupling g(X) greater than or similar to 10(-2) and gauge boson mediators X with M-X less than or similar to 10 MeV lead to large matter potential suppressing the sterile neutrino production before the neutrino decoupling. With this choice of parameter ranges, big-bang nucleosynthesis is left unchanged and gives no bound on the model. However, we show that at lower temperatures when active-sterile oscillations are no longer matter suppressed, sterile neutrinos are still in a collisional regime, due to their secret self-interactions. The interplay between vacuum oscillations and collisions leads to a scattering-induced decoherent production of sterile neutrinos with a fast rate. This process is responsible for a flavor equilibration among the different neutrino species. We explore the effect of this large sterile neutrino population on cosmological observables. We find that a signature of strong secret interactions would be a reduction of the effective number of neutrinos N-eff at matter radiation equality down to 2.7. Moreover, for M-X greater than or similar to g(X) MeV sterile neutrinos would be free-streaming before becoming nonrelativistic and they would affect the large-scale structure power spectrum. As a consequence, for this range of parameters we find a tension of an eV mass sterile state with cosmological neutrino mass bounds.