So, even today, both the Catholic and Orthodox Churches agree that the Roman Pope is the Protos, the first among all the other bishops. This is evident in the Ravenna Document, produced by representatives of the Catholic and Orthodox Churches gathered in the Joint International Commission for the Theological Dialogue between the Roman Catholic Church and the Orthodox Church on 13 October 2007.^(*) Where the disagreement emerges is in what this primacy actually means. In the traditional Orthodox understanding, the primacy of the Roman Pope is really just a position of honour, a primus inter pares among the other bishops. Meanwhile, the traditional Catholic understanding is that the primacy of the Roman Pope gives him supreme power and authority over all the other bishops in the Church.

As for how the different Churches interacted with each other before the [convenient] year 1054, u/Libertat wrote an excellent comment explaining the growth of Roman pontifical power vis à vis the other Patriarchs and bishops. Basically, the power dynamic between the Pope of Rome and the four Patriarchs in the East shifted and changed across the centuries. It was not unheard of for bishops and Patriarchs in the east to appeal to the Roman pontiff in order to leverage his prestige in local disputes. At the same time, however, from the 6th-8th centuries CE, the Emperors in Constantinople and their representatives, the Exarchs of Ravenna, exercised inordinate influence over the Roman Patriarchate, effectively controlling the election of Popes for nearly two centuries. Overall, if the four eastern Patriarchs (most importantly the Patriarch of Constantinople) had the support of the Emperor, they were effectively more powerful than the Protos, the Roman Pope. But, rarely did the eastern Patriarchs exercise any political influence within the Roman church (that would be left to the defender of the Church, the isapostolos Roman Emperor in Constantinople).

I hope this answer is satisfactory!

* Note, the Moscow Patriarchate was, to put it mildly, not pleased with the signing of the Ravenna Document. Muscovite attitudes towards ecumenism, however, are a discussion for another time.

EDIT: Formatting

The eastern church did not recognize the pope as head of the faith, they recognized the Roman emperor as the head of the faith, after the last vestiges of the eastern Roman Empire fell in 1453 with the fall of Constantinople, the head of the Eastern Orthodox Church passed to the tsar of Muscovy which later would became the Russian empire who continued to head the church until the fall of the Russian empire in the early 20th century

The power dynamic was before the split, they barely tolerated each other, the pope and Roman emperor(Byzantine, the word itself basically slander by the western church Bc they refused to acknowledge the eastern Roman Empire as the Roman Empire, which is also where we get the basis of the myth of the “dark ages” after “Rome collapsed” that 18th and 19th poets, writers and playwrights would create to essentially Pat themselves on the back Bc “of how they’ve come from their forefathers, how better they are, and how civilized they were) were constantly trying to get the other to recognize the other as head of Christendom and god’s representative on earth, and it continued this way until a the split where the pope excommunicated the eastern Roman emperor and all those who recognized him as head of the faith and labeled them heretics, the emperor in response did the same for the pope and all those who recognized him as head of the faith. After the relationship b/t the East and west was they ignored each other’s existence unless they had to, fought each other for various reasons, sometimes using religion as a shield and a legitimization for their goals, helping each when it suited the helper