This paper is devoted to the evaluation of various classical and more recent linearization schemes for nonlinear homogenization in terms of their efficiency to characterize local field fluctuations in nonlinear heterogeneous composites. It relies on an unbiased comparison between field statistics predicted by homogenization theories and those obtained from a reference solution solved using finite element techniques, based on the same microgeometry and boundary conditions and in which local nonlinear constitutive relations are exactly verified at each point. Two categories of linearization methods have been investigated: classical approaches based on a "stress-strain" approach (classical secant, classical and simplified affine) and methods based on "variational principles" (variational and Lahellec-Suquet procedures). For each approach, the maps and the statistical distribution functions of the local fields (strain, stress and incremental work) illustrating the intra- and inter-phase heterogeneities are provided for reinforced and porous power-law composites. This study supplements an earlier study focused on comparisons at the global level \cite{R06, R07} and provides additional information on the accuracy of some available classical and recent linearization procedures. The proposed methodology gives access to a deeper insight on nonlinear homogenization schemes and may eventually lead to improvements of these formulations.