Both one-proton and one-neutron knockout reactions were performed with fast beams of two asymmetric, neutron-deficient rare isotopes produced by projectile fragmentation. The reactions are used to probe the nucleon spectroscopic strengths at both the weakly and strongly bound nucleon Fermi surfaces. The one-proton knockout reactions Be-9(S-28, P-27)X and Be-9(Si-24, Al-23)X probe the weakly bound valence proton states and the one-neutron knockout reactions and Be-9(S-28, S-27)X and Be-9(Si-24, Si-23)X the strongly bound neutron states in the two systems. The spectroscopic strengths are extracted from the measured cross sections by comparisons with an eikonal reaction theory. The reduction of the experimentally deduced spectroscopic strengths, relative to the predictions of shell-model calculations, is of order 0.8-0.9 in the removal of weakly bound protons and 0.3-0.4 in the knockout of the strongly bound neutrons. These results support previous studies at the extremes of nuclear binding and provide further evidence that in asymmetric nuclear systems the nucleons of the deficient species, at the more-bound Fermi surface are more strongly correlated than those of the more weakly bound excess species.