Left ear advantages for tonal stimuli and music, indicating prevalently right hemisphere processing, and conversely, right ear advantages for speech, indicating left hemisphere processing, have been found repeatedly, from birth on. The monaural capacity to attend to and process speech or music may be useful in understanding the asymmetric processing strategies also in bilateral cochlear implant (CI) users. Namely, regardless of the reported benefits of early binaural implantation, children with CIs experience overall poorer speech perception in noise, and poorer music perception. While toddlers with bilateral CIs exhibit increased attention to speech, indexed by increased pupil dilation, only in the absence of significant background noise, no study has explored the monaural capacities to attend to and process speech and music in CI users, nor adults nor children. The present study explores pupillary behavior of 12 early bilaterally implanted congenitally deaf toddlers (1.5-4-y-old) while they were passively listening to speech (rhymed verses) or instrumental music at a constant intensity level (60 dB SPL), in quiet or with a 10 or 0 dB SNR babble noise, in an ecological environment (Ambisonics semi-sphere), with only the left or only the right cochlear implants switched on. The analysis of variance for the linear mixed model fitter by REML shows significant interaction between the stimulus (music, speech) and listening ear (left, right), irrespective of the background noise level. The overall increase in pupil dilation for music in right ear listening, and for speech in left ear listening, may indicate increased listening effort when listening to the respective stimuli with the “non-advantageous” ear. These asymmetries in pupil response may in turn reflect the hemispheric asymmetries in processing speech and music in early bilaterally implanted children, indicating that their rehabilitation protocols could take advantage of processing differences between the two ears.