The smart grid vision is to build an intelligent power network with an unprecedented level of situational awareness and controllability over its services and infrastructure. This paper advocates statistical inference methods to robustify power monitoring tasks against the outlier effects owing to faulty readings and malicious attacks, as well as against missing data due to privacy concerns and communication errors. In this context, a novel load cleansing and imputation scheme is developed leveraging the low intrinsic-dimensionality of spatiotemporal load profiles and the sparse nature of 'bad data.' A robust estimator based on principal components pursuit (PCP) is adopted, which effects a twofold sparsity-promoting regularization through an ℓ1-norm of the outliers, and the nuclear norm of the nominal load profiles. After recasting the non-separable nuclear norm into a form amenable to distributed optimization, a distributed (D-) PCP algorithm is developed to carry out the imputation and cleansing tasks using a network of interconnected smart meters. Computer simulations and tests with real load curve data corroborate the convergence and effectiveness of the novel D-PCP algorithm.