Although nonhuman primates' gestural communication is often considered to be a likely precursor of human language, the intentional properties in this communicative system have not yet been entirely elucidated. In particular, little is known about the intentional nature of monkeys' gestural signalling and related social understanding. We investigated whether olive baboons can (1) adjust their requesting gestures to the visual attention of the experimenter with special emphasis on the state of the eyes (open versus closed), and (2) flexibly tailor visual and auditory-based gestures to elaborate their communication as a function of whether or not the experimenter can see them. Using a food-requesting paradigm, we found monkeys able to favour either visual or auditory-based requesting gestures to match the experimenter's visual attention. Crucially, when the human was not visually attending, they silenced visual gestures to some extent but performed more attention-getting gestures. This is, to our knowledge, the first report of monkeys elaborating attention-getting signals to compensate for communication breakdown. Gestural communication was also supported by gaze alternation between the experimenter's face and the food, especially when the human was visually attending. These findings offer evidence that olive baboons understand the state of the eyes in others' visual attention and use requesting gestures intentionally. They emphasize that Old World monkeys shift to acoustic communication when the recipient is not visually attending. In contrast to that of human infants and great apes, this acoustic communication is purely gestural, not vocal.