The connection between the electronic polarizability and the decrease of the system size from macroscopic solid to nanoscale clusters has been addressed in a combined experimental and model-calculation study. A beam of free neutral potassium chloride clusters has been probed using synchrotron-radiation-based photoelectron spectroscopy. The introduction of "effective" polarizability for chlorine, lower than that in molecules and dimers and decreasing with increasing coordination, has allowed us to significantly improve the agreement between the experimental electron binding energies and the electrostatic model predictions. Using the calculated site-specific binding energies, we have been able to assign the spectral details of the cluster response to the ionizing X-ray radiation, and to explain its change with cluster size. From our assignments we find that the higher-coordination face-atom responses in the K 3p spectra increase significantly with increasing cluster size relative to that of the edge atoms. The reasons behind the decrease of polarizability predicted earlier by ab initio calculations are discussed in terms of the limited mobility of the electron clouds caused by the interaction with the neighboring ions.