We take advantage of the site-selective nature of the ^{75}As and ^{63}Cu NMR techniques to probe the Cu substitution effects on the local magnetic properties of the FeAs planes in Ba(Fe_{1-x}Cu_{x})_{2}As_{2}. We show that the suppression of antiferromagnetic Fe spin fluctuations induced by Cu substitution is weaker than a naive expectation based on a simple rigid band picture, in which each Cu atom would donate three electrons to the FeAs planes. Comparison between ^{63}Cu and ^{75}As NMR data indicates that spin fluctuations are suppressed at the Cu and their neighboring Fe sites in the tetragonal phase, suggesting the strongly local nature of the Cu substitution effects. We attribute the absence of a large superconducting dome in the phase diagram of Ba(Fe_{1-x}Cu_{x})_{2}As_{2} to the emergence of a nearly magnetically ordered FeAs plane under the presence of orthorhombic distortion.