New experiments that measure the low-frequency shot-noise spectrum at local tunneling contacts on mesoscopic structures are proposed. The current fluctuation spectrum at a single tunneling tip is determined by local partial densities of states. The current-correlation spectrum between two tunneling tips is sensitive to non-diagonal density of states elements which are expressed in terms of products of scattering states of the conductor. Thus such an experiment permits to investigate correlations of electronic wave functions. We present specific results for a clean wire with a single barrier and for metallic diffusive conductors.