Many problems in solid and geomechanics require the concept of a meso-continuum, which allows a resolution of stress and other dependent fields over scales not infinitely larger than the typical microscale. Passage from the microstructure to such a meso-continuum is based on a scale dependent window playing the role of a Representative Volume Element (RVE). It turns out that the material properties at the mesoscale cannot be uniquely approximated by a random field of stiffness with continuous realizations, but, rather, two random continuum fields, corresponding to essential and natural boundary conditions on RVE, need to be introduced to bound the material response from above and from below. In this paper Monte Carlo simulations are used to obtain the first- and second-order one- and two-point characteristics of these two random fields for random chessboards and matrix-inclusion composites. Special focus is on the correlation functions describing the autocovariances and crosscovariances of effective random meso-scale conductivity tensor Cij and its dual Sij . Following issues are investigated: i) scale-dependence of noise-to-signal ratios of various components of Cij and Sij , ii) spatial structure of the correlation function, iii) uniform strain versus exact calculations in determination of the correlation function, iv) correlation structure of composites with inclusions without and with overlap.