Current fluctuations provide information on the transport process which is not contained in the conductance. For example the effective charge q_eff describing shot noise yields information on the charge transferred and also on the energy-dependent correlations between charge transfers. We study nonequilibrium noise correlations in diffusive normal-superconducting structures in the presence of a supercurrent. We adopt the full counting statistic approach and present a parametrization for the quasiclassical Green's function in the first order of the counting field χ. This we employ to obtain the voltage and phase dependence of cross and autocorrelations and to describe the role played by the setup geometry. We find that the low-voltage behavior of q_eff is a result of a competition between anticorrelation of Andreev pairs due to proximity effect and the depression of the local density of states. Furthermore, we show that the noise correlations are independent of the sign of the supercurrent.