Lossy bosonic channels play an important role in a number of quantum information tasks, since they well approximate thermal dissipation in an experiment. Here, we characterize their metrological power in the idler-free and entanglement-assisted cases, using respectively single- and two-mode Gaussian states as probes. In the problem of estimating the loss parameter, we study the power-constrained quantum Fisher information (QFI) for generic temperature and loss parameter regimes, showing qualitative behaviours of the optimal probes. We show semi-analytically that the two-mode squeezed-vacuum state optimizes the QFI for any value of the loss parameter and temperature. We discuss the optimization of the {\it total} QFI, where the number of probes is allowed to vary by keeping the total power constrained. In this context, we elucidate the role of the ``shadow-effect'', or passive signature, for reaching a quantum advantage. Finally, we discuss the implications of our results for the quantum illumination and quantum reading protocols.