The performance of gold nanoparticles (NPs) in applications depends critically on the structure of the NP-solvent interface, at which the electrostatic surface polarization is one of the key characteristics that affects hydration, ionic adsorption, and electrochemical reactions. Here, we demonstrate significant effects of explicit metal polarizability on the solvation and electrostatic properties of bare gold NPs in aqueous electrolyte solutions of sodium salts of various anions (Cl$^-$, BF$_4$$^-$, PF$_6$$^-$, Nip$^-$(nitrophenolate), and 3- and 4-valent hexacyanoferrate (HCF)), using classical molecular dynamics simulations with a polarizable core-shell model of the gold atoms. We find considerable spatial heterogeneity of the polarization and electrostatic potentials on the NP surface, mediated by a highly facet-dependent structuring of the interfacial water molecules. Moreover, ion-specific, facet-dependent ion adsorption leads to large alterations of the interfacial polarization. Compared to non-polarizable NPs, polarizability modifies water local dipole densities only slightly, but has substantial effects on the electrostatic surface potentials, and leads to significant lateral redistributions of ions on the NP surface. Besides, interfacial polarization effects on the individual monovalent ions cancel out in the far field, and effective Debye-Huckel surface potentials remain essentially unaffected, as anticipated from continuum `image-charge concepts. Hence, the explicit charge response of metal NPs is crucial for the accurate description and interpretation of interfacial electrostatics (as, e.g., for charge transfer and interface polarization in catalysis and electrochemistry).