Studies on auditory laterality have revealed asymmetries for processing, particularly species-specific signals, in vertebrates and that each hemisphere may process different features according to their functional “value”. Processing of novel, intense emotion-inducing or finer individual features may require attention and we hypothesised that the “functional pertinence” of the stimuli may be modulating attentional processes and hence lateralisation of sound processing. Behavioural measures in “(food) distracted” captive Campbell’s monkeys and electrophysiological recordings in anesthetised (versus awake) European starlings were performed during the broadcast of auditory stimuli with different functional “saliences” (e.g., familiar/novel). In Campbell’s monkeys, only novel sounds elicited lateralised responses, with a right hemisphere preference. Unfamiliar sounds elicited more head movements, reflecting enhanced attention, whereas familiar (usual in the home environment) sounds elicited few responses, and thus might not be arousing enough to stimulate attention. In starlings, in field L, when awake, individual identity was processed more in the right hemisphere, whereas, when anaesthetised, the left hemisphere was more involved in processing potentially socially meaningless sounds. These results suggest that the attention-getting property of stimuli may be an adapted concept for explaining hemispheric auditory specialisation. An attention-based model may reconcile the different existing hypotheses of a Right Hemisphere-arousal/intensity or individual based lateralisation.