Experiments that measure the average current and current fluctuations at one or two local tunnelling contacts on a mesoscopic multiprobe conductor are proposed and theoretically investigated. The average current and the current fluctuations at a single tunnelling tip are determined by an effective local distribution function, which is given as the product of local partial densities of states (injectivities) and the Fermi distribution functions in the electron reservoirs. The current correlations at two tips connected to a phase-coherent conductor are determined by spatially non-diagonal density of states elements and can give information about the phase and correlations of wavefunctions. All results are illustrated for measurements on a ballistic wire and on a metallic diffusive wire.