Lagrangian quantization of the free superfield gauge theories of higher massless superspins is performed both in anti-de Sitter and flat superspaces. For the models under consideration the straightforward application of the BV procedure for quantization of reducible theories leads to immense calculations. In order to avoid the difficulty, a simplification of this procedure in reduction coordinates is considered. The contribution to the effective action is shown to be independent of the gauge structure of the classical formulations dually equivalent to each other. It is the same both for actions with finite and infinite stages of reducibility.