Memristors are expected to be one of the key building blocks for the development of new bio-inspired nanoelectronics. Memristive effects in transition metal oxides are usually linked to the electromigration at the nanoscale of charged oxygen vacancies (OV). In this paper we address, for Pt/TiOx/TaOy/Pt devices, the exchange of OV between the device and the environment upon the application of electrical stress. From a combination of experiments and theoretical simulations we determine that both TiOx and TaOy layers oxidize, via environmental oxygen uptake, during the electroforming process. Once the memristive effect is stabilized (post-forming behavior) our results suggest that oxygen exchange with the environment is suppressed and the OV dynamics that drives the memristive behavior is restricted to an internal electromigration between TiOx and TaOy layers. Our work provides relevant information for the design of reliable binary oxide memristive devices.