  We formulate a microscopic theory of non-local electron transport in
three-terminal diffusive normal-superconducting-normal (NSN) structures with
arbitrary interface transmissions. At low energies $\varepsilon$ we predict
strong enhancement of non-local spectral conductance $g_{12} \propto
1/\varepsilon$ due to quantum interference of electrons in disordered
N-terminals. In contrast, non-local resistance $R_{12}$ remains smooth at small
$\varepsilon$ and, furthermore, is found to depend neither on parameters of NS
interfaces nor on those of N-terminals. At higher temperatures $R_{12}$
exhibits a peak caused by the trade-off between charge imbalance and Andreev
reflection. Our results are in a good agreement with recent experimental
observations and can be used for quantitative analysis of future experiments.
