  We work out a unified theory describing both non-local electron transport and
cross-correlated shot noise in a three-terminal normal-superconducting-normal
(NSN) hybrid nanostructure. We describe noise cross correlations both for
subgap and overgap bias voltages and for arbitrary distribution of channel
transmissions in NS contacts. We specifically address a physically important
situation of diffusive contacts and demonstrate a non-trivial behavior of
non-local shot noise exhibiting both positive and negative cross correlations
depending on the bias voltages. For this case, we derive a relatively simple
analytical expression for cross-correlated noise power which contains only
experimentally accessible parameters.
