We investigate how a stochastic gravitational-wave background, produced from a discrete set of astrophysical sources, differs from an idealized model consisting of an isotropic, unpolarized, and Gaussian background. We focus, in particular, on the different signatures produced from these two cases, as observed in a cross-correlation search. We show that averaged over many realizations of an astrophysical background, the cross-correlation measurement of an astrophysical background is identical to that of an idealized background. However, any one realization of an astrophysical background can produce a different signature. Using a model consisting of an ensemble of binary neutron star coalescences, we quantify the typical difference between the signal from individual realizations of the astrophysical background and the idealized case. For advanced detectors, we find that, using a cross-correlation analysis, astrophysical backgrounds from many discrete sources are probably indistinguishable from an idealized background.