We report, through 17O NMR, an unambiguous local determination of the intrinsic kagome lattice spin susceptibility as well as that created around nonmagnetic defects arising from natural Zn/Cu exchange in the S=1/2 (Cu2+) herbertsmithite ZnCu3(OH)6Cl2 compound. The issue of a singlet-triplet gap is addressed. The magnetic response around a defect is found to markedly differ from that observed in nonfrustrated antiferromagnets. Finally, we discuss our relaxation measurements in the light of Cu and Cl NMR data and suggest a flat q dependence of the excitations.