Specifically post-Theodosius when the split became permanent. Also, what name would one have use to describe the other?
I can't really answer this question at all, but I do have two minor comments;
The only way to approach this question would be through primary sources reflecting the opinions of a broad range of both communities, of which I am pretty sure there are none. That said, you could start with Procopius and Theophanes and see what comes up. IIRC, there'll be something about racing factions and religious mania - not much, but, would indicate that even common people might have had strongly divided opinions over some basic issues like religious practice. This is not quite East vs West, I know.
There are a lot of questions like this on askhistorians these days - like this in the sense that very modern concepts and identities are being applied to questions where they might not be appropriate. For example, your question depends somewhat on a concept of national identity that is a little out of place in the 4th century CE - not that people who were citizens didn't think of themselves as 'Roman', but that the idea of a nation-state with distinct and discrete boundaries that defined its inhabitants national identities is not quite what was going on. I'll have to return to this point later as I am now surrounded by tiny people clamouring for their breakfast.
Look at some earlier discussions on the topic: http://www.reddit.com/r/AskHistorians/comments/2cftvq/how_distinct_was_the_western_and_eastern_roman/
Basically, the split was considered to be an administrative division of the single Empire, rather than two distinct countries as we sometimes think of it today.