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For any Bell locality scenario (or Kochen-Specker noncontextuality scenario), the joint Shannon entropies of local (or noncontextual) models define a convex cone for which the nontrivial facets are tight entropic Bell (or contextuality) inequalities. In this paper we explore this entropic approach and derive tight entropic inequalities for various scenarios. One advantage of entropic inequalities is that they easily adapt to situations such as bilocality scenarios, which have additional independence requirements that are nonlinear on the level of probabilities, but linear on the level of entropies. Another advantage is that, despite the nonlinearity, taking detection inefficiencies into account turns out to be very simple. When joint measurements are conducted by a single detector only, the detector efficiency for witnessing quantum contextuality can be arbitrarily low.