This is a really complicated question on one level, if we are to get too deep into the various ideological constructs that surrounded the Roman notion of an "Empire" was. But there is at least one polity that I can point to some direct attestations of as being described as vast and powerful (if not in all other respects admirable): what we today call the Achaemenid Empire.
Plutarch's Life of Artaxerxes:
6.1 Nevertheless, restless and factious men thought that affairs demanded Cyrus, a man who had a brilliant spirit, surpassing skill in war, and great love for his friends; and that the magnitude of the empire required a king of lofty purpose and ambition.
20.1 Nevertheless, he could not capture them, but though they had lost Cyrus their leader and their own commanders, they rescued themselves from his very palace, as one might say, thus proving clearly to the p175 world that the empire of the Persians and their king abounded in gold and luxury and women, but in all else was an empty vaunt.
28.3 However, it was the greatness of the empire and the fear which Dareius felt towards Ochus that paved the way for Teribazus although, since Aspasia had been taken away, the Cyprus-born goddess of love was not altogether without influence in the case.
(The term translated "empire" here is basileia)
These discussions - including Plutarchs' moralistic remarks on the vauntness of the polity beyond its vast wealth - tie into something I have previously argued. Namely, that there seems to be a thread starting in the Classical era and carried on through authors like Plutarch and Arrian, that sees the ancient kings of Persia as highly emblematic of the institution of monarchy. You can see this among other things in the frequent usage of titles such as simply "the King" (ho basileus) or "the Great King" (ho megas basileus) rather than "the King of Persia" even in circumstances where it would not necessarily be unambiguous, something that I know is remarked on in the Landmark edition of Arrian's Anabasis. There's a sort of preoccupation with the quaintness of court ritual and intrigue (real or otherwise) evident from Ctesias' Persica (4th c BC) and carried on by authors like Diodoros and Plutarch. Here's one example of Diodorus emphasizing the vast reach of the King's dominion:
22.1 King Artaxerxes had learned some time before from Pharnabazus that Cyrus was secretly collecting an army to lead against him, and when he now learned that he was on the march, he summoned his armaments from every place to Ecbatana in Media. 2 When the contingents from the Indians and certain other peoples were delayed because of the remoteness of those regions, he set out to meet Cyrus with the army that had been assembled
And in another part of his World History, Diodoros discusses the court intrigue of Persia:
17.5.3 As our narrative is now to treat of the kingdom of the Persians, we must go back a little to pick up the thread. While Philip was still king, Ochus15 ruled the Persians and oppressed his subjects cruelly and harshly. Since his savage disposition made him hated, the chiliarch Bagoas, a eunuch in physical fact but a militant rogue in disposition, killed him by poison administered by a certain physician and placed upon the throne the youngest of his sons, Arses...
The story of Bagoas seems to have been popular, and it suggests to me an idea of how these authors viewed Persia - it was vast and powerful, but still "weak" in that it was subject to the whims and intrigues of its decadent, effete court life. This particular narrative ends with the appointment of Dareios III, the last Great King and one with a much weaker claim to the throne than his predecessors, who ends up turning the tables on Bagoas and killing him. On a side note, Arrian cites a supposed letter by Alexander claiming that in fact Dareios III had had his brief predecessor Artaxerxes IV murdered, which is another example of the predominant relevance these anecdotes had in the minds of the authors who kept them alive (as opposed to mundane matters of administration and regular rule in the dominions subject to the Persian monarch, which they appear to have been pretty uninterested in!)
Unfortunately, there are only a few major stories that get repeated over and over with slight variations by Roman-era authors (mainly the life of Cyrus the Great, Xerxes' invasion, Cyrus the Younger's rebellion, Alexander's invasion), so we only get a limited view, and it is not really clear to me how Roman subjects would relate the Kingdom of the Persians to the polity they themselves lived under. However, even from these few examples, we have some idea: these authors clearly possessed an awareness of the Persian Empire as an incredibly vast and powerful dominion that had existed well before Rome's rise to prominence in later centuries, and still regarded the tales of its court intrigue and the character of its rulers as relevant and instructive topics of discussion.