In scenarios such as remote work, open offices and call centers, multiple people may simultaneously have independent spoken interactions with their devices in the same room. The speech of competing speakers will however be picked up by all microphones, both reducing the quality of audio and exposing speakers to breaches in privacy. We propose a cooperative cross-talk cancellation solution breaking the single active speaker assumption employed by most telecommunication systems. The proposed method applies source separation on the microphone signals of independent devices, to extract the dominant speaker in each device. It is realized using a localization estimator based on a deep neural network, followed by a time-frequency mask to separate the target speech from the interfering one at each time-frequency unit referring to its orientation. By experimental evaluation, we confirm that the proposed method effectively reduces crosstalk and exceeds the baseline expectation maximization method by 10 dB in terms of interference rejection. This performance makes the proposed method a viable solution for cross-talk cancellation in near-field conditions, thus protecting the privacy of external speakers in the same acoustic space.