We present a theory for the current shot noise in long diffusive SNS structures with low-resistive interfaces at arbitrary temperatures. In such structures, the noise is mostly generated by normal electron scattering in the N-region. Whereas the $I$-$V$ characteristics are approximately described by Ohms law, the current noise reveals all characteristic features of the MAR regime: giant enhancement at low voltages, pronounced SGS, and excess noise at large voltages. The most spectacular feature of the noise in the incoherent MAR regime is a universal finite noise level at zero voltage and at zero temperature, $S= 4Delta/3R$. This effect can be understood as the result of the enhancement of the effective charge of the carriers, $q^{it eff}=2Delta/V$, or, alternatively, as the effect of strongly non-equilibrium quasiparticle population in the energy gap region with the effective temperature $T_0=Delta/3$. Under the condition of dominant electron-electron scattering, the junction undergoes crossover to the hot electron regime, with the effective temperature of the subgap electrons decreasing logarithmically with the voltage. Calculation of the noise power has been done on the basis of circuit theory of the incoherent MAR.
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