The use of multichannel measurements is a current practice for source characterization in multiple fields. But common to all experimental approaches is the presence of extraneous noise such as calibration, electronic or ambient noise. However, signals are supposed to be stationary and performing averaging of cross-spectral quantities over several time snapshots will concentrate uncorrelated noise along the cross-spectral matrix (CSM) diagonal. A common practice is thus to set the CSM diagonal to zero, which is known to improve the dynamic range of the source localization maps, yet this also leads to underestimated source levels. More advanced techniques have been recently developed to avoid such problems by preserving or reconstructing source information that lies in the CSM diagonal. Several existing approaches for CSM denoising are investigated in this paper and a new one is proposed as well. We consider an unknown number of uncorrelated sources and no reference background noise. The proposed method is based on the decomposition of the CSM into a low-rank part and a residual diagonal part attached to the unwanted noise; the corresponding inference problem is set up within a probabilistic framework which is ideally suited to take the non-deterministic nature of the estimated CSM into account. This is then solved by computing the maximum a posteriori estimates of the decomposition by estimating the full a posteriori prob- ability distribution by running a Markov chain Monte Carlo. For each method, reconstruction errors are compared in the frame of various numerical experiments, for different acoustic signals and noise structures.