Improved membranes and improved membrane processes for treating gas streams containing hydrogen sulfide and methane, plus water vapor, carbon dioxide or both. The processes rely on the availability of two membrane types, one of which has a high hydrogen sulfide/methane selectivity and a high water vapor/methane selectivity, when measured with multicomponent gas mixtures at high pressures. Based on the different permeation properties of the two membrane types, optimized separation processes can be designed. In favorable cases, the processes can simultaneously dehydrate the gas stream and remove the hydrogen sulfide to very low levels.