A strict analytical solution of the problem of spin-noise signal formation in a volume medium with randomly moving spin carriers is presented. The treatment is carried out in the model of light scattering in a medium with fluctuating inhomogeneity. Along with conventional single-beam geometry, we consider the two-beam arrangement, with the scattering field of the auxiliary (tilted) beam heterodyned on the photodetector illuminated by the main beam. It is shown that the spin-noise signal detected in the two-beam arrangement is highly sensitive to motion (diffusion) of the spin carriers within the illuminated volume and thus can provide additional information about the spin dynamics and spatial correlations of spin polarization in the volume media. Our quantitative estimates show that, under real experimental conditions, spin diffusion may strongly suppress the spin-noise signal in the two-beam geometry. The mechanism of this suppression is similar to that of the time-of-flight broadening with the critical distance determined by the period of two-beam spatial interference rather than by the beam diameter.