It's not a dumb question because the split was mostly administrative and not an official breakup of the Empire.

They did go to civil war against each other but not in the sense they were two distinct entities and were vying for supremacy, but more so the old fashioned civil war in Ancient Roman history, an Emperor controlled the West with another controlling the East and they had disputations of one sort or another.

An example of when they went to war with each other was when Eugenius became Western Roman Emperor around 392 AD. The Eastern Empire suspected that the previous Western Roman Emperor Valentinian II was assassinated and the real power behind the throne was a Frankish general, as such, the Eastern Empire viewed Eugenius as a usurper. The war also had some other underpinnings such as the Italian Roman aristocracy in the West was still mostly Pagan, while the Eastern Emperor was Christian.

They went to war which actually lead to the Eastern Roman Emperor, Theodosius, becoming the last Roman Emperor to rule over a unified Roman Empire for under a year before his death.