We determine long-term correlations in the time series of fluctuating x-ray speckle intensities. A fluctuation analysis of small angle x-ray scattering data of the two phase-separating alloys Al-6 at. %Ag and Al-9 at. %Zn at late stages of phase separation reveals long-term correlations that are dramatically different for the two systems. From a comparison with recent Monte Carlo simulations we conclude that two different coarsening mechanisms are predominant in the two alloys---coarsening either by diffusion of single atoms or by movement of whole precipitates.