The nature of the way that this system emerged as well as how the Romans themselves viewed the divide offers insight into this question. To them it wasn't such a clear cut divide it was more like a split of administrative duties which emerged out of the Crisis of the 3rd Century (a period where there were a series of threats such as plague, invasion and civil war) as the Emperor could not react to all such threats at once - especially the military ones as Rome faced opposition from multiple sources all over it's borders. The precursor to the divide was the Tetrarchy instigated by Diocletian in 293 where there would be a senior Emperor (Augustus) in the the Western half and Eastern half and a junior one (Caesar) to support him leading to 4 Emperors total. So there was no border dispute as what simply had happened was a divide of administrative duties. Indeed this is how the Romans themselves viewed it: not as two separate empires but as one whole one both being referred to as the Roman Empire (indeed the people of the later Byzantine Empire never stopped calling themselves the Roman Empire or Romans.) Therefore, there was no major dispute over the borders even later on when the divide became more solidified as at the time they were still one Empire working together, indeed in general there was mostly cooperation between the two halves of the Empire with sometimes siblings ruling both West and East or being joined by marriage alliances. Later on, however, during the West's dying days it became generally accepted that the Western Emperor was a more junior Emperor and would deffer to the Easter Emperor.

The borders themselves also had precedence in history in the 2nd Triumvirate with Mark Anthony being in control of roughly the same area as the later divide in the East and Octavian being likewise in the West - it seemed, to the Romans, at least that the divide should be between the Adriatic. It also makes sense as in terms of Geography the areas are of a roughly equivalent size.

It could also be argued that there was perhaps a cultural difference between the two halves of the empires. In the West you have Italy and the Romanised provinces compared to Greece and the predominantly Hellenised provinces (the Successor Kingdoms forged out of the remnants of Alexander the Great's Empire.) The divide could therefore be said to have come from this.