Goldstone's theorem states that in a relativistic quantum field theory, spontaneous breaking of an exact continuous global internal symmetry produces a massless Goldstone boson for each independent broken generator. More precisely, a conserved global current whose charge acts nontrivially on the vacuum expectation value has a massless pole in an appropriate current-field correlation function. The usual counting assumes a Lorentz invariant vacuum state and the standard locality and positivity hypotheses.
A local gauge symmetry is a gauge redundancy, so it does not by itself imply massless physical Goldstone bosons. In the Higgs mechanism, the corresponding scalar directions provide the longitudinal polarizations of massive gauge bosons.

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