The important thing to remember here is that the Roman Empire was never officially split into two different states. According to official state ideology and in the minds of its citizens the empire was always a united entity, even if at times it was governed by more than one emperor. The terms ‘Western Roman Empire’ and ‘Eastern Roman Empire’ should be understood as administrative units of a unitary state, because that is how the Romans themselves viewed it. This means that, at least ideally, the different emperors were always meant to work together as a harmonious governing body, even if they were individually responsible for different geographical parts of the empire. This becomes obvious when you look at Late Roman laws. They are usually issued in the name of the entire council of emperors. Take for example the famous edict ‘cunctos populos’ of 380 AD, which made Nicene Christianity the only official doctrine of the empire. It starts like this:
EMPERORS GRATIAN, VALENTINIAN AND THEODOSIUS AUGUSTI. EDICT TO THE PEOPLE OF CONSTANTINOPLE.
It is our desire that all the various nations which are subject to our Clemency and Moderation [...]
The edict is directed towards the people of Constantinople, a city in the eastern part of the empire, at the time ruled over by Theodosius I, while his co-emperors Gratian and Valentinian II were governing the west. Nevertheless all three reigning emperors are invoked at the start of the edict.
As neither the Western nor the Eastern Roman Empire were separate sovereign entities they couldn’t just independently choose their own rulers. As long as there still was at least one legitimate ruling emperor he had to be involved somehow. Ideally this meant that he would simply pick his co-ruler himself, normally a member of his close family like a son or a brother. More rarely someone unrelated would be chosen, like when Gratian and Valentinian II made Theodosius ruler of the east after their uncle Valens had died in battle.
In principle it was also possible to retroactively recognize someone who had independently risen to the throne. For example when in 306 AD Constantine I was acclaimed Augustus by his troops after the death of his father the other emperors pragmatically recognized him, at least as Caesar, even though this violated the principles of the tetrarchy. But this was a dangerous precedent to set as it could easily open the door for further usurpations. Because of that the usual answer to such independent claims to power was civil war. In this answer I described just such a case. After the death of Honorius, who had ruled the west, his nephew Theodosius II in the east refused to choose a successor, maybe because he planed to rule the whole empire by himself. The elites of the Western Empire however by now were used to be governed by their own emperor and so after some waiting they picked their own candidate, a senior civil servant called John. Constantinople saw this as a usurpation and in a swift civil war replaced John with Theodosius’ cousin Valentinian III.
Now let’s jump to the year 800 AD. When pope Leo III crowned Charlemagne emperor at Christmas of that year neither one of them had sought the approval of the eastern court. It was only afterwards that the Frankish ruler was trying to gain the recognition of their deed. And Constantinople unsurprisingly was in no mood to give it. The crowning had been a huge affront to the sovereignty of the ruling empress Irene. From the Byzantine point of view she would have been the only one capable of choosing another emperor. And even if she would have been inclined to do such a thing, Charlemagne would have been a more than unlikely choice. In Byzantine eyes he didn’t even qualify as a Roman. He wasn’t borne inside the empire, had never served it and didn’t even speak Greek. Also, he clearly was the ruler of a foreign power, the kingdom of the Franks. His foreignness categorically disqualified him from any claim to the Roman Empire. So to the Byzantines Charlemagne’s coronation was not only a usurpation, it was the usurpation of a Barbarian. More than enough reason to be outraged.