We construct holomorphic functional calculi and introduce local quadratic estimates for operators in a reflexive Banach space that are bisectorial except possibly in a neighbourhood of the origin. The main result is an equivalence of local quadratic estimates with bounded holomorphic functional calculi. For operators with spectrum in a neighbourhood of the origin, the results are weaker than those for bisectorial operators. For operators with a spectral gap in a neighbourhood of the origin, the results are stronger. In each case, however, local quadratic estimates are a more appropriate tool than standard quadratic estimates for establishing that our functional calculi are bounded. This shows that in certain applications it suffices to establish local quadratic estimates.