At the time the Western Roman Empire fell, the legion system was long defunct. The legionary system required a high level of social order and efficient taxation to support the infrastructure that made it so effective in the first place. A declining population, westward movement of warlike immigrant tribes, a fall in trade and the excises and duties and the goods they transported, the debasement of the currency, the erosion of traditional Roman social values and institutions all contributed to the disintegration of the Western Roman Empire.

As a result, traditional recruiting incentives such as the promise of land, steady pay, a chance at loot and slaves, and citizenship were either becoming not worthwhile or not promising. Additionally, a proper legion requires drillmasters, blacksmiths, engineers, tanners, quartermasters, and a whole slew of noncombatant, highly specialized skills. As trade and prosperity dropped off, so did the degree of specialization in society.

In the end, the W. Roman Empire incorporated the invading barbarians into their army. Foederati swore an oath to protect the borders of the Empire in exchange for land, but as you can imagine, they were none too loyal and possessed nowhere near the quality, discipline, or fidelity of the original legionaries. They predictably turned against their masters and those sworn to protect the Empire actually helped produce its downfall. Look at the late emperors and observe how several of them aren't even Romans ethnically.

To answer your question, Rome's soldiers did indeed desert the Empire, but not to Byzantium.

This is a bit tricky to answer. Desertions obviously happened and I'm sure many soldiers did join up with Eastern Roman forces, but this is hard to quantify and no obvious source comes to mind. A brief search through the likely secondary literature (Heather, Goldsworthy, Halsall and MacGeorge) didn't find anything either. However, we do know a bit about how various political/military leaders in the West allied themselves with the East, but this wasn't outright defection - in all the cases I can think of, they were using their Eastern Roman connections to support their own interests in the last days of the Western Roman Empire.

The most obvious candidate was a man named Marcellinus, who was effectively a Roman general-turned-warlord in Dalmatia. He was for example deeply involved in Eastern Roman efforts to take back North Africa for the West and his nephew was Julius Nepos, the man imposed by the East as the second-to-last (or the last) emperor of the West. Marcellinus however had no interest in deserting wholesale to the East - the historian Priscus alluded to a planned attack on Italy, which was perhaps an invasion with the goal of removing Ricimer, the 'barbarian' generalissimo who was governing the empire through various puppet emperors, and placing himself on the throne. Marcellinus was allegedly persuaded to do otherwise by Constantinople, but it is still nonetheless clear that he was an independent agent rather than an Eastern Roman puppet. In an age of declining centralised authority, it made sense for powerful individuals to seize power for themselves - ideally to make a bid for the empire itself, if not, well, seizing a province or two wasn't too shabby either. Placing themselves under the authority of another distant emperor can hardly have appealed to these ambitious men.

It is also worth noting that Constantinople at this point wanted to stabilise the West, not to accelerate its end. Money, soldiers and candidates for the throne were repeatedly thrown into Italy, so Constantinople certainly would not have encouraged mass defections to the East. Obviously, in times of conflict between the two empire it would have been encouraged, but it was not the norm and these conflicts were never about fundamental differences, but only happened when the East's interests were threatened. Generally, I think it would be fair to say that Constantinople wanted Western soldiers to stay where they were and defend their empire.

Now to address /u/MeaterBeaterEater's comment, which provides a rather old-fashioned perspective on the 'fall' of Rome.

At the time the Western Roman Empire fell, the legion system was long defunct. The legionary system required a high level of social order and efficient taxation to support the infrastructure that made it so effective in the first place. A declining population, westward movement of warlike immigrant tribes, a fall in trade and the excises and duties and the goods they transported, the debasement of the currency, the erosion of traditional Roman social values and institutions all contributed to the disintegration of the Western Roman Empire.

The legion system had long been defunct, but the forces in the empire's disposal were still very powerful indeed. After all, it was through this transformed Roman army (which had changed constantly in Roman history) that men like Constantine, Julian, Valentinian and Theodosius were able to win so many crushing victories over the 'barbarians'. This changed in the fifth century, but modern historians generally blame either the pressure caused by mass migrations or internal Roman disputes for Roman military failures in this period, not the structural 'inferiority' of the army. The army, as always, was effective under strong leaders, and less successful when the leadership was divided or incompetent.

In the end, the W. Roman Empire incorporated the invading barbarians into their army. Foederati swore an oath to protect the borders of the Empire in exchange for land, but as you can imagine, they were none too loyal and possessed nowhere near the quality, discipline, or fidelity of the original legionaries. They predictably turned against their masters and those sworn to protect the Empire actually helped produce its downfall. Look at the late emperors and observe how several of them aren't even Romans ethnically.

Many 'barbarians' were perfectly loyal to the empire. Men liked Ricimer and Gundobad may have played a kingmaker role and deposed emperors whom they didn't like, but they were still keen on defending the empire itself. The empire was rich, influential and gave them power - why would they abandon it? These men also sought to defend their own personal interests, which was hardly unusual. In any case, the original legionaries were definitely not entirely loyal to the Roman state, as we only have to look at the number of generals who marched on Rome in the days of the Republic and early Empire. 'Barbarian' leaders certainly played a role in the 'fall' of the West, but it wasn't because they were disloyal. Instead, it was their personal conflicts and power struggles with other powerful individuals that drained the dwindling resources of the empire, which makes them no different to all the 'proper' Romans who made plays for power at the same time. Even 'barbarian' peoples well beyond Roman control at this point, such as the Visigoths and the Vandals, rarely wanted to the empire to fall - the empire was just another player (or tool, given how weak it was) in the complex world of late-antique diplomacy, destroying it was therefore often contradictory to these peoples' interests. In any case, Roman Italy lived on under the protection of the 'barbarians' after 476, and the 'fall' itself was not seen as a particularly dramatic moment for contemporaries.