We present theory for a current shot noise in mesoscopic diffusive superconductor-normal metal-superconductor (SNS) junctions. The current shot noise is tremendously enhanced at small applied voltage, V < Δ/e, due to the mechanism of multiple Andreev reflection (MAR), which creates long correlated trains of transmitted electrons. The central result for the MAR regime is a universal value for the noise power in the limit of small voltage and at zero physical temperature, which corresponds to the effective noise temperature of the order of superconducting energy gap Δ. At very small voltage, MAR is destroyed by inelastic relaxation, consequently, the level of noise reduces and approaches at V = 0 the thermal noise value. We investigate the suppression of MAR by inelastic scattering processes and analyze the crossover from the MAR regime to a hot electron regime.