In short, they are not. But allow me to elaborate.

While 1054 is the date commonly thrown around for the Great Schism, it was not recognized at the time as creating a permanent split. There are several reasons for this. Perhaps most importantly, the culmination of the events of 1054 was the mutual excommunication of Cardinal Humbert, the papal legate, and Michael Kerularios, the Patriarch of Constantinople. This excommunication, however, was not understood at the time to extend to a complete separation between the Latin and Greek Churches. Rather, it was understood on each side to be an excommunication of a particular person. While you can't excommunicate the Patriarch of Constantinople and not expect it to affect your relationship with the Church of Constantinople, the idea on neither side was to separate the two Churches. Furthermore, Pope Leo IX, the pope in whose stead Cardinal Humbert was acting, had actually died before the excommunication was issued (though Humbert didn't know this), which put the legal status of the excommunication in serious doubt.

Besides the personal nature of the excommunications of 1054, there is also the fact that these events were hardly unprecedented. There had been periods during which Rome and Constantinople had been out of communion with each other before, perhaps most notable in the early 6th century during the Acacian Schism and the 9th century during the Photian Schism. In all of these cases Rome and Constantinople had eventually sorted out their issues and returned to harmony. Even if the participants in the events of 1054 had understood there to be a rupture between the two Churches, they would have had every reason to suspect that it would be resolved in the not-too-distant future. In fact, in the 1080's Byzantine Emperor Alexios Komnenos asked why the Greek Church was not in communion with the Latin Church? After a review of the chancellory, the ecclesiastical authorities determined that there was no reason, and Alexios began a process of trying to heal a schism of whose origin no one was sure. This never came to fruition because of the perhaps the defining event of East-West ecclesial relations for this period: the Crusades.

The Crusades are where the Schism hardened from idea into reality. Before the Crusades, Rome and Constantinople were largely content to leave each other alone, with certain geographic exceptions (South Italy, the Balkans, etc.) as well as temporal ones (the Photian Schism, 1054, etc.). This changed with the Crusades, as Latin Christians now frequently found themselves in immediate contact with Byzantine Christians. This contact served to emphasize for both sides just how different the two traditions had become. This mutual incomprehension can be seen in the newly-minted Byzantine genre of "Lists of the Errors of the Latins." These lists emphasized ritual differences between the Latins and Greeks, especially focusing on the seeming barbarism of the Latins. On the canonical level, a schism is defined as a situation where two bishops each claim jurisdiction over a particular see. This first happened for East and West when in 1098 the Crusaders took the city of Antioch, one of the 5 major sees of Christendom. After severe mistreatment by the Crusaders, the Greek Patriarch, John the Oxite, fled the city. The Crusaders replaced him with a Latin Patriarch. There were now to bishops, one Latin and one Greek, each of whom claimed jurisdiction over Antioch. This situation was soon repeated in Jerusalem, and eventually in Constantinople in 1204.

This date, 1204, can be understood in some sense as the "real" date of the Schism. When the Crusaders of the 4th Crusade sacked Constantinople, one could say that the Rubicon of ecclesial relations was crossed. Before this a reconciliation between Rome and Constantinople still seemed plausible if difficult. Afterwards, the Byzantine hostility towards Latin Christians reached a new level of hostility that makes ecclesial reunion difficult to imagine. I won't go into it here (though if you're interested I could try to elaborate in a comment), but developments in papal self-understanding also meant that by the early 13th century, the Popes were seriously disinclined to make any concessions to their power to try to bring the Greek Church back into the fold, so to speak. After 1204 the Greeks saw the Latins as an invading force and, not unrelatedly, a religious "other" and the schism that had wavered somewhere between fiction and reality during the 12th century became an incontrovertible fact. Schism exists largely where it is believed to exist, and it is only the events of 1204 that caused the schism to take such firm root in the minds and hearts of each side that there was no way back as there had been during previous ecclesiastical spats, such as the aforementioned Acacian and Photian Schisms. So, to get back to your question, almost nothing really changed for a Byzantine bishop in the immediate aftermath of 1054. Even for centuries later, nobody would point to the date 1054 as the origin of the Schism. This date arose in later (which is to say early modern and 19th century) scholarship because, well, people wanted a decisive date for something that really happened not at one particular time or another but over the course of 150 years of more. I hope that helps, and please ask any questions either that arose from my answer or that I didn't adequately address.