I mean the Roman Empire was massive, so I can't imagine that a regular person outside of the Italian peninsula would even know anything had changed for weeks or months, right?
And on top of that, how did the lives of the average person in the WRE and ERE change when they (respectively) dissolved?
I assume by division into two empires you mean the period between AD 395 and AD 476 (or AD 480) when the Empire was ruled by two emperors: one in the East and one in the West. Note that it was still the single empire during that time, not only theoretically but in the practice as well: the same constitutional laws applied to the both realms, the military cooperated, emperors in the west looked for recognition from their senior colleagues in Constantinople, etc.
But to answer your question - nothing really changed in 395 when Theodosius I died and his sons Arcadius and Honorius took titles of the Emperor of the East and West. Theodosius himself was the Emperor of the East except for the final few months of his life when he ruled both halves of the Empire. The institution of imperial colleagues was very familiar to Romans - even the first Emperor Augustus took Tiberius as a co-emperor to ensure the smooth transition of power.