Hard X-ray Photoemission spectroscopy (PES) of copper core electronic states, with a probing depth of $sim$60 AA, is used to show that the Zhang-Rice singlet feature is present in La$_2$CuO$_4$ but is absent in Nd$_2$CuO$_4$. Hole- and electron doping in La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) and Nd$_{2-x}$Ce$_x$CuO$_4$ (NCCO) result in new well-screened features which are missing in soft X-ray PES. Impurity Anderson model calculations establish metallic screening as its origin, which is strongly suppressed within 15 $text{AA}$ of the surface. Complemented with X-ray absorption spectroscopy, the small chemical-potential shift in core levels ($sim0.2$ eV) are shown to be consistent with modifications of valence and conduction band states spanning the band gap ($sim1$ eV) upon hole- and electron-doping in LSCO and NCCO.
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