I understand many of our sources come from Ancient greek writers and historians, and therefore much of what we know is from what the Greeks saw. But why is it that peoples on the Eastern side of the empire haven't recorded similar histories?
Lack of pre-Achaemenid writing extant at the time. In later eras, we find that the writing used is derived from Greek or Aramaic alphabets - for example, the first examples of writing in Indic languages are recorded in the Edicts of Ashoka in the 2nd century BC, in an Aramaic-derived script. The Bactrian language of the Kushans was written in Greek script, and the Parthian Script is more or less plain Aramaic. The golden age of written culture in India, for example, is during the Gupta Empire in the 3rd to 6th centuries AD, and it is only in the late Sasanian and early Islamic era that the silk road cities Central Asia become intellectual and scholarly strongholds.
Simply put, there was, as far as we can tell, no tradition of writing (or otherwise recording) "histories" the way there was such a burgeoning tradition in Greece. Oral transmission focused on religious hymns, epic, and traditional wisdom.