Decoherence in quantum searches, and in the Grover search, in particular, has already been extensively studied, leading very quickly to the loss of the quadratic speedup over the classical case, when searching for some target (marked) element within a set of size N. The noise models used were, however, almost always global. In this paper, we study Grover search under the influence of localized partially dephasing noise of rate p. We find that, in the case when the size k of the affected subspace is much smaller than N and the target is unaffected by the noise, namely when kp≪√N, the quadratic speedup is retained. Once these restrictions are not met, the quadratic speedup is lost. If the target is affected by the noise, the noise rate needs to scale as 1/√N to keep the speedup. We also observe an intermediate region, where if k∼N^μ and the target is unaffected, the speedup seems to obey N^μ, which for μ>0.5 is worse than the quantum, but better than the classical case. We also put obtained results for quantum searches into perspective of quantum walks and searches on graphs.