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We consider speech enhancement for signals picked up in one noisy environment that must be rendered to a listener in another noisy environment. For both far-end noise reduction and near-end listening enhancement, it has been shown that excessive focus on noise suppression or intelligibility maximization may lead to excessive speech distortions and quality degradations in favorable noise conditions, where intelligibility is already at ceiling level. Recently [1,2] propose to remedy this with a minimum processing framework that either reduces noise or enhances listening a minimum amount given that a certain intelligibility criterion is still satisfied Additionally, it has been shown that joint consideration of both environments improves speech enhancement performance. In this paper, we formulate a joint far- and near-end minimum processing framework, that improves intelligibility while limiting speech distortions in favorable noise conditions. We provide closed-form solutions to specific boundary scenarios and investigate performance for the general case using numerical optimization. We also show concatenating existing minimum processing far- and near-end enhancement methods preserves the effects of the initial methods. Results show that the joint optimization can further improve performance compared to the concatenated approach.