  Injection of unpolarized charge current through the longitudinal leads of a
four-terminal two-dimensional electron gas with the Rashba spin-orbit (SO)
coupling and/or SO scattering off extrinsic impurities is responsible not only
for the pure spin Hall current in the transverse leads, but also for random
time-dependent current fluctuations. We employ the scattering approach to
current-current correlations in multiterminal nanoscale conductors to analyze
the shot noise of transverse pure spin Hall and zero charge current, or
transverse spin current and non-zero charge Hall current, driven by unpolarized
or spin-polarized longitudinal current, respectively. Since any spin-flip acts
as an additional source of noise, we argue that these shot noises offer a
unique tool to differentiate between intrinsic and extrinsic SO mechanisms
underlying the spin Hall effect in paramagnetic devices.
