If a Buddha exists then can a private buddha arise or exist somewhere in the same world system without having learned from the current Buddha? The question is in the title. 

Or is he just known as an arahant?
I don't remember where the story is mentioned, but if I remember right, there was a Pacceka Buddha in the world when the Bodhisatta was born as prince Siddhartha. He attained Parnibbana after hearing the news from the Devas that the Buddha to be has appeared in the world. So it would seem that a Pacceka Buddha wouldn't stay if a SammaSambuddha appears in the world.  Also, those who are capable of attaining enlightenment will be drawn towards the Buddha or the Buddha will see them when he scans the world system every morning.