A fitting system is configured to determine a first power consumption metric for a first hearing device included in a binaural hearing system configured to wirelessly communicate with an external device during a communication session, determine a second power consumption metric for a second hearing device included in the binaural hearing system, determine that the first power consumption metric is lower than the second power consumption metric, and assign, based on the determination that the first power consumption metric is lower than the second power consumption metric, a first operating role to the first hearing device. The first operating role specifies that the first hearing device is to be wirelessly connected with the external device during the communication session.