  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 \gtrsim 10^{-2} and gauge boson mediators $X$ with $M_X \lesssim 10$ 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 $N_{\rm eff}$ at matter radiation equality down
to 2.7. Moreover, for $M_X \gtrsim g_X$ MeV sterile neutrinos would be
free-streaming before becoming non-relativistic and they would affect the
large-scale structure power spectrum. As a consequence, for this range of
parameters we find a tension of a eV mass sterile state with cosmological
neutrino mass bounds.
