In the present simulation work we study graphene-based polymer nanocomposites composed of hydrogenated and carboxylated graphene sheets dispersed in polar and nonpolar short polymer matrices (i.e., matrices :containing chains with low molecular weight). The aim of our work is to examine spatial and dynamical heterogeneities of such systems and to provide a compact picture about the effects of the edge group functionalitation Of graphene sheets on the properties of hybrid graphene-based materials. We perform atomistic molecular dynamics simulations of edge-functionalized graphene sheets embedded in poly(ethylene oxide) (polar matrix) and polyethylene (nonpolar matrix). We :choose a low loading of the graphene nanofiller (from 1.7% to 3.6%) in agreement with experimental data. We further implement a detailed analysis of static and dynamic properties of polymer chains on the level of both the entire hybrid material and the polymer/graphene interface through a new approach that is able to distinguish between the adsorbed and edge region around the nanofiller. At the local stale, strong structural and dynamical heterogeneities are observed; i.e., the behavior of the polymer matrix appears to he highly affected by the presence of the edge-functionalized graphene. Slow dynamics was detected in the adsorbed: layers in all nanocomposites. Additionaly, interaction of grafted carboxylated groups with polar matrix led to a further delay in the segmental as well as the chain relaxation close to the graphene edges. This effect seems to slow down the chain dynamics even more than the actual adsorption of the chain on the surface. Overall average orientational dynamics of polymer chains in nanocomposites as well as the collective dynamics quantified through the single chain coherent structure factor reveal slight deviations from the bulk behavior in the terminal region, presumably due to the small percentage of the slow dynamic component at the given loading of the nanofiller. Enhancement of the rheological properties is not Observed within the time window of our simulations. Our results emphasize the importance of the surface/polymer interactions in the graphene-based nanocomposites and suggest that by A proper choice of edge-grafted groups we can achieve better material performance.