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This paper is concerned with a communication channel with two senders and one receiver, in which each sender observes a private feedback signal. The two feedback signals are not necessarily equivalent to or derived from the signal observed by the receiver. An achievable rate region is demonstrated for this multiple-access channel by means of a new superposition coding scheme. In particular it is shown that a rate region, previously obtained for the multiple-access channel with "perfect" feedback to both senders, remains achievable when the feedback connection to one of the senders is eliminated.