A microscopic theory of spin fluctuations in an ensemble of electrons localized on donors in a bulk semiconductor has been developed. Both the hyperfine interaction of the electron spin with spins of host lattice nuclei and the exchange interaction between the electrons have been taken into account. A model of clusters to calculate spin noise spectra of the ensemble of localized charge carriers has been proposed. It has been shown that the electron-electron exchange interaction leads to an effective averaging of random nuclear fields and a shift of the peak in the spin-fluctuation spectrum towards lower frequencies.