We report an x-ray emission spectroscopy study of the local fluctuating magnetic moment (μ_bare) in NaFe_1-xCo_xAs and NaFe_1-xCu_xAs. In NaFeAs, the reduced height of the As ions induces a local magnetic moment higher than BaFe₂As₂ despite lower T_N and ordered magnetic moment. As NaFeAs is doped with Co, μ_bare is slightly reduced, whereas Cu doping leaves it unaffected, indicating a different doping mechanism: based on electron counting for Co, whereas impurity scattering dominates in the case of Cu. Finally, we observe an increase in μ_bare with temperature in all samples as observed in electron- and hole-doped BaFe₂As₂. Since both Co and Cu doping display superconductivity, our findings demonstrate that the formation of Cooper pairs is not connected with the complete loss of fluctuating paramagnetic moments.