Label-free bottom-up proteomics using mass spectrometry and liquid chromatography has long been established as one of the most popular high-throughput analysis workflows for proteome characterization. However, it produces data hindered by complex and heterogeneous missing values, which imputation has long remained problematic. To cope with this, we introduce Pirat, an algorithm that harnesses this challenge using an original likelihood maximization strategy. Notably, it models the instrument limit by learning a global censoring mechanism from the data available. Moreover, it estimates the covariance matrix between enzymatic cleavage products (ie peptides or precursor ions), while offering a natural way to integrate complementary transcriptomic information when multi-omic assays are available. Our benchmarking on several datasets covering a variety of experimental designs (number of samples, acquisition mode, missingness patterns, etc.) and using a variety of metrics (differential analysis ground truth or imputation errors) shows that Pirat outperforms all pre-existing imputation methods. Beyond the interest of Pirat as an imputation tool, these results pinpoint the need for a paradigm change in proteomics imputation, as most pre-existing strategies could be boosted by incorporating similar models to account for the instrument censorship or for the correlation structures, either grounded to the analytical pipeline or arising from a multi-omic approach.