Entanglement distillation is a fundamental information processing task whose implementation is key to quantum communication and modular quantum computing. Noise experienced by such communication and computing platforms occurs not only in the form of Pauli noise such as dephasing (sometimes called $T_2$) but also non-Pauli noise such as amplitude damping (sometimes called $T_1$). We initiate a study of practical and asymptotic distillation over what we call the joint damping-dephasing noise channel. In the practical setting, we propose a distillation scheme that completely isolates away the damping noise. In the asymptotic setting we derive lower bounds on the entanglement sharing capacities including the coherent and reverse coherent information. Like the protocol achieving the reverse coherent information, our scheme uses backward only communication. However for realistic damping noise ($T_1 \neq 2T_2$) our strategy can exceed the reverse coherent strategy which is the best known for pure damping. In addition, our companion paper [arXiv:2405.06231] presents evidence showing that the channel displays non-additivity at the 2-letter level.