There's certainly no real analogue to the historians you described. There's really not anything description of the Greeks at all. The most descriptive line is probably the tomb inscription of Darius the Great calling Macedonians "shield-hat wearing Greeks," in a list of peoples and places within the empire. If you look just after the Persian period, the Hellenistic Babylonian priest Berossus, almost qualifies. However, his work is known only through a few scattered fragments and was written as a history of Babylonia to be consumed by a Greek-speaking audience. It's not really a text that we'd expect to see an outside perspective on Greeks even if we had the whole thing.

The Achaemenid Persians and the other peoples of the imperial core didn't really have historians in the same sense that the Greeks were developing at the same time. The genre of detailed, researched, comparative, narrative record keeping that is all-but synonymous with history today had to be invented. That process was actively occurring (for West Asia, Europe, and North Africa) during the Achaemenid period, but that development was largely confined to the eastern Mediterranean at first. Obviously there are lots of well preserved Greek historians who we used to generally define the genre. There were also writers cited by later Greco-Roman writers Mochus of Sidon and Xanthus of Lydia whose works are now lost. There are even aspects of the early historical genre in the Bible and the pseudepigraphic work from Second Temple Judaism.

The closest we can really get to a historian writing from the Persian heartland is Ctesias of Cnidus, a Greek physician in the court of Artaxerxes II who wrote a history of Persia in the 4th Century BCE. Unfortunately, much of Ctesias' work is undercut by the fact that he seems to have preferred legends and dramatic rumors over more realistic stories of Persian history whenever possible.

The Persians themselves had no equivalent genre. Strictly speaking, the Persians themselves barely had written tradition of any sort at this point. The first written document in the Persian language was probably an early draft of Darius the Great's Behistun Inscription, which may have been the impetus to invent a Persian script in the first place. Up to this point, and for generations going forward, most of Persian and Iranian history, religion, mythology, etc was remembered as an oral tradition. From the documents that survives, it seems the Babylonian chroniclers are the closest we have to a written history of events during the Persian period, simply in a summary of major events year by year. Even then, we've only found the chronicles immediately predating Cyrus the Great's conquest of Babylon, an excerpt from Artaxerxes III, and a section from immediately after Alexander the Great's conquests.

As for Rome, there's no evidence that the Achaemenids would have written anything about the the Romans even if they were writing history. There is very little evidence for Achaemenid interaction west of Greece. The only narrative account of Persians attempting to learn more about Italy comes from Herodotus, who describes a Persian exploratory expedition around 500 BCE as the farthest west any Persian had ever gone. On that expedition, their Greek guide abandoned them as soon as he was close to his hometown and the Persians were forced to turn back and follow the coast home.