We study the impact of the environment on the evolution of galaxies in the zCOSMOS 10 k sample in the redshift range 0.1 ≤ z ≤ 1.0 over an area of ~1.5 deg2. The considered sample of secure spectroscopic redshifts contains about 8500 galaxies, with their stellar masses estimated by SED fitting of the multiwavelength optical to near-infrared (NIR) photometry. The evolution of the galaxy stellar mass function (GSMF) in high and low density regions provides a tool to study the mass assembly evolution in different environments; moreover, the contributions to the GSMF from different galaxy types, as defined by their SEDs and their morphologies, can be quantified. At redshift z ~ 1, the GSMF is only slightly dependent on environment, but at lower redshifts the shapes of the GSMFs in high- and low-density environments become extremely different, with high density regions exhibiting a marked bimodality, not reproducible by a single Schechter function. As a result of this analysis, we infer that galaxy evolution depends on both the stellar mass and the environment, the latter setting the probability of a galaxy to have a given mass: all the galaxy properties related to the stellar mass show a dependence on environment, reflecting the difference observed in the mass functions. The shapes of the GSMFs of early- and late-type galaxies are almost identical for the extremes of the density contrast we consider, ranging from isolated galaxies to rich group members. The evolution toward z = 0 of the transition mass M_cross, i.e., the mass at which the early- and late-type GSMFs match each other, is more rapid in high density environments, because of a difference in the evolution of the normalisation of GSMFs compared to the total one in the considered environment. The same result is found by studying the relative contributions of different galaxy types, implying that there is a more rapid evolution in overdense regions, in particular for intermediate stellar masses. The rate of evolution is different for sets of galaxy types divided on the basis of their SEDs or their morphologies, tentatively suggesting that the migration from the blue cloud to the red sequence occurs on a shorter timescale than the transformation from disc-like morphologies to ellipticals. Our analysis suggests that environmental mechanisms of galaxy transformation start to be more effective at z < 1. The comparison of the observed GSMFs to the same quantities derived from a set of mock catalogues based on semi-analytical models shows disagreement, in both low and high density environments: in particular, blue galaxies in sparse environments are overproduced in the semi-analytical models at intermediate and high masses, because of a deficit of star formation suppression, while at z < 0.5 an excess of red galaxies is present in dense environments at intermediate and low masses, because of the overquenching of satellites. Based on observations obtained at the European Southern Observatory(ESO) Very Large Telescope (VLT), Paranal, Chile, as part of the Large Program 175.A-0839 (the zCOSMOS Spectroscopic Redshift Survey).