You can breath through either or both. In short which ever is most comfortable for you to maintain a moderate breath rate. Both LiveO2 and ExtremeO2 work regardless of how you choose to breathe.
If the mask is fitted properly you should have no difficulty breathing in through your nose. If this is difficult, move the mask higher on the bridge of you nose, inflate the cushion a little more and or loosen the neoprene straps slightly.
Most individuals will breath through their mouth when breathing hard but this is not necessarily the most efficient way to breathe.
Yoga suggests the best breathing method is to inhale through the nose with the tongue on the roof of the mouth and then exhale through an opened mouth with the tongue flat and relaxed. However, this can be challenging when exercising vigorously. During exertion, breathing through the mouth is typical, especially with high altitude, like with ExtremeO2. Just breath any way that feels good. The main objective here is to bring the oxygen from the system into your body.