High-energy neutrinos are expected to be produced in a variety of astrophysical sources as well as in optically thick hidden sources. We explore the matter-induced oscillation effects on emitted neutrino fluxes of three different flavors from the latter class. We use the ratio of electron and tau induced showers to muon tracks, in upcoming neutrino telescopes, as the principal observable in our analysis. This ratio depends on the neutrino energy, density profile of the sources, and on the oscillation parameters. The largely unknown flux normalization drops out of our calculation and only affects the statistics. For the current knowledge of the oscillation parameters we find that the matter-induced effects are non-negligible and the enhancement of the ratio from its vacuum value takes place in an energy range where the neutrino telescopes are the most sensitive. Quantifying the effect would be useful to learn about the astrophysics of the sources as well as the oscillation parameters. If the neutrino telescopes mostly detect diffuse neutrinos without identifying their sources, then any deviation of the measured flux ratios from the vacuum expectation values would be most naturally explained by a large population of hidden sources for which matter-induced neutrino oscillation effects are important.