Wideband power spectrum sensing is fundamental for numerous applications. When side information on the potentially active emitters is available, such as carriers and spectral masks, it should be exploited to improve sensing performance. Here the power spectrum is modeled as a weighted sum of candidate spectral density primitives. The objective is to estimate the unknown weights from a few randomly filtered broadband power measurement bits, taken using a network of low-end sensors. A linear programming formulation that exploits the sparsity in the unknown weights is proposed. A better approach follows, which exploits the approximately Gaussian distribution of the errors in the power measurements prior to quantization, in a maximum likelihood formulation that includes a sparsity-inducing penalty term. Simulations show that the model weights can be accurately estimated from few bits, even when the errors are significant.