Correlations and other collective phenomena in a schematic model of heterogeneous binary agents (individual spin-glass samples) are considered on the complete graph and also on 2d and 3d regular lattices. The system's stochastic dynamics is studied by numerical simulations. The dynamics is so slow that one can meaningfully speak of quasi-equilibrium states. Performing measurements of correlations in such a quasi-equilibrium state we find that they are random both as to their sign and absolute value, but on average they fall off very slowly with distance in all instances that we have studied. This means that the system is essentially non-local, small changes at one end may have a strong impact at the other. Correlations and other local quantities are extremely sensitive to the boundary conditions all across the system, although this sensitivity disappears upon averaging over the samples or partially averaging over the agents. The strong, random correlations tend to organize a large fraction of the agents into strongly correlated clusters that act together. If we think about this model as a distant metaphor of economic agents or bank networks, the systemic risk implications of this tendency are clear: any impact on even a single strongly correlated agent will spread, in an unforeseeable manner, to the whole system via the strong random correlations.