How did the enemies of Rome view the empire? Did they understand it's scale/strength ?

There's lots of ways to answer this, because the (Western) Roman Kingdom/Republic/Empire lasted from 753 BC until around AD 476, and had many different enemies, and experienced fluctuating fortunes and levels of dominance in different places at different times.

So I'm going to pick a period, though still a broad one. That will be the Empire in the period of its greatest power, and its greatest challenges: The Principate (from the reign of Augustus) until things started to go irreparably bad in the 300s AD. The two best topics for my answer would be the two greatest enemies of the Roman empire, the Germanic peoples across the Rhine and Danube, and the Parthian/Sasanid Empire. Alas, I will pick just one, or I won't be able to cover what I want to say in a reasonably sized post. So I will pick the Germanic peoples.

A general issue with answering this: we have no literary sources from the Germans, who had no such culture I am aware of (or even the Parthians/Sasanids, who were very well written but whose literary tradition was lost in the upheavals of the Muslim conquests and later developments). So I won't be giving you simple, unequivocal answers from the mouths of those the question concerns, but rather answers borne of a bit of educated guesswork (based on reading lots of source material over many years!).

In general, the Roman Empire relied upon their reputation, and the impression of overwhelming force, to deter its enemies, as its territory was enormous and considering its size the army was remarkably small. For comparison, the Nazis kept hundreds of thousands of soldiers in Norway throughout all of WWII, whereas the entire Roman Empire had something like 300-400,000ish military personnel (including sailors) for the whole Empire for most of the Imperial period. Of course, the Romans could drop ten legions or so on the head of anyone that they really needed to, like Tiberius brought against Maroboduus in Dacia (Velleius 2.113), or Germanicus against Arminius (Tacitus Annals 2.16). But as a rule, Rome relied upon its reputation and the impression of overwhelming force to deter potential aggressors. For more on this see Mattern, esp. 83, 108, 119, 121-2.

Rome had established such a reputation and military hegemony in the time of Augustus that it no longer had any serious enemies. While individuals such as Boudicca and Arminius made sometimes successful attacks in order to reclaim their own territory, these were no serious threat to the Empire and other than Maroboduus, whom Augustus crushed via. his very capable general and future emperor Tiberius, Rome's central territories could not be threatened by any Germanic or British tribal enemy. This lack of existential threat to Rome did not change from the destruction of the Cimbri and Teutones by Gaius Marius in 104 BC until around 160 AD under Marcus Aurelius, when larger and more powerful than ever before Germanic tribes, the Marcomanni and the Quadi, went on the offensive (Austin and Rankov 205). This was the Marcomannic Wars, in which the Imperial army was outright defeated c. AD 170, and both Italy and Greece itself were directly assaulted. From this period onward, the threat of Germanic groups escalated, the threat being further enhanced by the development of tribal confederations, huge groupings of already powerful tribes under names such as the Alamanni, that could directly challenge the Romans for military supremacy (Austin and Rankov 211). The fact that the Germans were become wedged between the westward movements of the incredibly formidable steppe peoples and the military might of the Rhine/Danube legions must have been a decisive factor in this development, though what caused the steppe migrations themselves is not decisively known (Austin and Rankov 210).

However, the Roman Empire was still respected because its army was still the best in the world in open battle, despite the increasing sophistication of Germanic armies. It was only Adrianople in AD 378 that permanently shattered the impression of Roman military superiority (Austin and Rankov 243). After this point, the Roman Empire was increasingly assailed by emboldened Germanic tribes until it collapsed. The Roman Army remained numerous and formidable even in this terrible period, as evidenced by decisive victories such as Strasbourg in 357 and at the Catalaunian Plains as late as 451. But the illusion of its invincibility in Europe was gone, and so the Roman Army, in the grand sense always outnumbered by its potential enemies, was overwhelmed and ceased to exist when central government collapsed. The dire legacy of Adrianople for Rome was not the material losses to the Empire, which were significant, but the fatal blow to the prestige of Rome.

Modern studies on the Roman Army's tactics and strategic superiority (most especially Goldsworthy and Mattern) emphasise the psychological superiority of the Roman Army against its enemies. At a tactical level, very few would dare to fight the legions in open battle, and those that did generally regretted it in short order. This must have been well known. Only after a very long period and under great pressure from all sides did the Germanic peoples unite and utilise their incredible potential military power to overcome the Rhine and Danube defences, but it still took a prominent blow such as Adrianople to open the door for the Germans who were waiting just across the river.

The power of Rome was well understood in this period: the most brazen attacks on Rome by Germans had either limited aims (Arminius, who wanted to avoid direct combat with the legions whenever possible), were borne of desperation (the Goths at Adrianople, perhaps also the revolt of Vercingetorix), or were just miscalculations or perhaps emotional decisions (Boudicca). This phase of Roman dominance was shaken by aggression by the Marcomanni, Quadi, and later groups such as the Alamanni, but Rome restored its physical dominance and its reputation once Diocletian wrapped up the Crisis of the Third Century.

So in answer to the question: I'm going to say that the Germans viewed Rome as a fixture in the world, the ultimate military power, until the prestige of Rome was shattered at Adrianople. After Adrianople, the Germanic tribes finally could perhaps for the first time ever have viewed a group such as the Huns as a deadlier foe than Rome. They all wanted to participate in it, many Germans served in its legions and many more benefited from trade and cultural exchange, and when the steppe migrations began pushing the Germans west, they wanted the pleasant climate and safety of Rome. Germans served in the Roman Army in great numbers from as early as the first century AD (eg. German auxiliaries in Judea under Titus), and they took great pride in it. So while Rome was no doubt an enemy to larger groups such as the Alamanni, who had no place in the Empire they could not carve out, to individual Germans it must been a place of awe-inspiring power and opportunity. This can be observed by the loyal service of the Franks and even the Goths until Roman mismanagement shattered that relationship. Great groups such as the Franks and very clearly the Goths wanted nothing more than to own their own little corner of the Empire: replacing it could not have been their intention until its weakness made such service no longer of use to them. There is great respect for the power of Rome in these relationships, even though groups such as the Goths were very formidable in their own right. The respect of the Germans for Rome's power was a significant element of what kept Rome, with its relatively small and overstretched professional army, on its feet for so long. The Romans never forgot how to fight, but in my view it was the collapse of this respect for the Roman Empire's size and strength that was a significant element (perhaps the most significant element?) in the fall of the Western Empire.

Key sources:

Austin, N.J.E. and Rankov, N.B., Exploratio: Military and Political Intelligence in the Roman World from the Second Punic War to the Battle of Adrianople, London: Routledge 1995

Goldsworthy, Adrian, The Roman Army At War; 100 BC- AD 200, Oxford: Clarendon Press, 1996.

Mattern, Susan P., Rome and the Enemy: Imperial Strategy in the Principate, Los Angeles: University of California Press, 1999

Bonus source: There is a really interesting article by Peter S. Wells (author of The Battle That Stopped Rome, which I don't agree with but I enjoyed anyway) titled "The Germanic and Danubian Transfrontier Peoples" in pages 589-600 of The Oxford Handbook of Warfare in the Classical World, Eds. Brian Campbell and Lawrence A. Tritle, Oxford University Press, New York, 2013. It doesn't deal with this issue exactly but it's a good article and if you can find it you should read it.