The present paper develops a collaborative scheme whereby cognitive radios cooperate to localize active primary transmitters and reconstruct the power spectral density (PSD) maps (one per frequency band) portraying the power distribution across space. The sensing scheme relies on a parsimonious linear system model that accounts for the narrow-band nature of transmit-PSDs compared to the large swath of sensed frequencies, and for the group sparsity emerging when adopting a spatial grid of candidate primary user locations. Combining the merits of Lasso, group Lasso, and total least-squares (TLS), the proposed group sparse (GS) TLS approach yields hierarchically-sparse PSD estimates, and copes with model uncertainty induced by channel randomness and grid mismatch effects. Taking advantage of a novel low-complexity solver for the GS-Lasso, a block coordinate descent scheme is developed to solve the formulated GS-TLS problem. Simulations demonstrate the superior localization and PSD-estimation performance of GS-TLS compared to approaches that do not account for model uncertainties.