  We consider a current-carrying, phase-coherent multi-probe conductor to which
a small tunneling contact is attached. We treat the conductor and the tunneling
contact as a phase-coherent entity and use a Green's function formulation of
the scattering approach. We show that the average current and the current
fluctuations at the tunneling contact are determined by an effective local
non-equilibrium distribution function. This function characterizes the
distribution of charge-carriers (or quasi-particles) inside the conductor. It
is an exact quantum-mechanical expression and contains the phase-coherence of
the particles via local partial densities of states, called injectivities. The
distribution function is analyzed for different systems in the zero-temperature
limit as well as at finite temperature. Furthermore, we investigate in detail
the correlations of the currents measured at two different contacts of a
four-probe sample, where two of the probes are only weakly coupled contacts. In
particular, we show that the correlations of the currents are at
zero-temperature given by spatially non-diagonal injectivities and
emissivities. These non-diagonal densities are sensitive to correlations of
wave functions and the phase of the wave functions. We consider ballistic
conductors and metallic diffusive conductors. We also analyze the Aharonov-Bohm
oscillations in the shot noise correlations of a conductor which in the absence
of the nano-contacts exhibits no flux-sensitivity in the conductance.
