Sensory laterality is influenced by the individual’s attentional state. There are variations in the way different individuals of a same species attend to stimuli. When confronted to novelty, some individuals are more explorative than others. Curiosity is composed of sensation and knowledge seeking in humans. In the present study, we hypothesized that more curious animals, i.e., showing more sensory exploration would be less lateralized than quietly attentive individuals, performing instead more gazing behaviours. In order to test this hypothesis and its possible generality, we performed two studies using two animal models (dolphins and starlings) and two modalities (visual and auditory) of presentation of species-specific and non-species-specific stimuli. Both dolphins and starlings presented more gazes for the species-specific stimuli and more exploratory components for the non-species-specific stimuli. Moreover, in both cases, the non-species-specific stimuli involved more lateralized responses whereas there was no or less clear laterality for the species-specific stimuli. The more exploratory dolphins and starlings also showed a decreased laterality: the more “curious” individuals showed no laterality. Further studies are needed on characterization of curiosity in relation to attention structure. The present study suggests that individual variations in sensory laterality may help disentangle the subtle differences between curiosity, attention and boldness.