Minimally.

First of all, the Visigoths and Ostrogoths should not be thought of as two halves of a shared people. The 'Goths' who lived north of the Danube in the old Roman provinces of Dacia in the 3rd through 6th centuries CE were an amalgamated cultural group to begin with. The primary noble houses and kinships may have been ethnically 'Gothic,' whatever that means, but the Gothic Peoples comprised numerous sub-ethnic groups including some of the formerly Roman inhabitants of Dacia. They were a polyglot people.

Secondly, by the time the first major group of Goths entered the Roman Empire (permanently) they were incredibly Romanized. Possibly more so than any other Northern European group in the period aside from the Franks. Numerous Goths served in Roman Armies of the late Empire, they were Christianized (Arian), etc. The archaeology of the period suggests that the Northern European culture groups north of the Eastern Danube in the late 4th century lived lives which were virtually identical, in terms of material culture, to those of the people on the South side of the Danube, except in comparitive wealth levels.

Thirdly, the first major group of 'Goths' that permanently moved into the Empire, whose descendants would eventually come to be known as the Visigoths, spent roughly 20-30 years (a whole generation) living inside the Empire and compositing with other collections of desperate peoples inside the Empire before Alaric finally sacked Rome. Landless Roman Military deserters, former slaves, other 'barbarians' and so on all flocked to Alaric's growing force as he pingponged back and forth through the Central Empire and into Italy. By the time the 'Visigoths' came to settle in Aquitaine and begin to build their kingdom, fully two generations of already highly Romanized 'Goths' had grown up inside of Roman civilization.

Foruth, the Ostrogoths had a very different story. Fleeing from the wrack of the Hunnic Empire, the leaders of one of the largest 'Gothic' groups to escape into the Empire after Atilla's death became one of the powerful freefloating military groups orbiting the court of the Eastern Roman Empire as the Western Empire slowly collapsed. Constantly in and out of favor, officially sanctioned as a component of the Roman military and then unsanctioned, the Ostrogoths came to be essentially a marching Roman Army in all but name and legitimacy at a time when the Eastern Empire had very few of those available for service in the Balkans. Many of the elite upper crust of the Ostrogoths received military training from Roman commanders and a Roman elite-level classical education from the top private tutors in Constantinople. At a time when a Romano-Isaurian General could become the Emperor Zeno, was much of the Ostrogoths were as Roman as a Roman could be, even in comparison to the elites of Italy and Constantinople. Theodoric himself was among them.

When Theodoric was dispatched to Italy to govern it on the Eastern Emperor's behalf, using his Romano-Ostrogothic army to fight the armies of Odoacer to a standstill before killing him, the Roman elites of Italy were far quicker to embrace the cultured, civilized Theodoric as their ruler than they had been to live under the military dominance of a rough soldier like Odoacer, and Theodoric made every effort to prove that he could match the best of the Italian natives in matters of culture and sophistication. There are even some suggestions that the Italians tried to elevate Theodoric from de facto military governor to junior Emperor, a move which Constantinople seemed to give a discreet nod towards by sending to Theodoric the imperial robes and diadem returned to Constantinople after Odoacer deposed the last Western Emperor. In his brief reunification of Southern Gaul, Illyria, Italy, and Hispana, Theodoric nearly seemed to BE the Western Emperor returned.

Now, for all this--there were some attempts made to elevate the 'Goths' above the rest of the Roman citizenry under both regimes. Much has been made of laws passed under Theodoric in Italy and by Visigothic kings of Toulouse prior that stipulated some additional legal privileges for Goths that Romans were not entitled to. It used to be thought that this suggested the 'ethnic' domination of one group over another, but in more recent years this notion has been reexamined, and the more recent thinking is that by this point, in both 'Gothic' realms, as well as in the emerging Frankish realm in the north, the supposed 'ethnic' marker had become more of a sort of 'warrior class' marker. Ethnic Romans who joined the armies of Theodoric come to be referred to and treated, under law, like Goths, despite their Roman Names and pedigrees. The idea being that these 'Gothic' groups were little more than armed military bands with sizeable camp followings, not fullscale ethnic groups migrating great distances. The armed military band of Alaric's Goths took possession of Southern France at Roman invitation and 'garrisoned' it more so than they resettled a large ethnic group there. On a slightly larger scale, Theodoric brought him military forces, his 'Goths,' to Italy, where they garrisoned it. Now this 'garrisoning' was different from the way Roman legions in the past had garrisoned parts of the Empire, but mostly in that the army commanders replaced the civilian administration which had previously existed there as the representatives of Imperial power. It was 'intended' to be temporary, especially as far as the Visigoths were concerned, and Eastern Roman policy towards Theodoric and his heirs fluctuated so much between Zeno, Anastasias, and Justin/Justinian that it's hard to tell if Theodoric had been intended to become the some sort of quasi-Imperial figure inside the Imperial college, or more of a military governor, or just to garrison Italy temporarily until Constantinople could come up with a more long term solution and then oop! four decades happened to go by.

Anyway, in the carving out of special privileges for warrior class members, these early Sub Roman Gothic rulers were both continuing a late Roman trend towards granting soldiers special legal privileges in exchange for service and laying the groundwork for the warrior aristocracy that would come to dominate the Early and High Middle Ages, but it wasn't a barbarisation: it was a logical, even foreseeable evolution of late Roman army trends once the central power in the region became less reliant on taxes gleaned from powerful civic landholders and more reliant on the increasingly voluntary military service and support of a permanent military hierarchy that began to buy, marry, and appropriate its way into the land holding elite of the territories they would not be permanently garrisoning. Something of a similar sort took place in Asia Minor and Thrace, in the Eastern Empire, after the Arab Conquests forced a necessitated transformation of the bulk of Imperial forces into permanently garrisoned local units who had to settle in the region and slowly merge with the local landowning classes to become a sort of landed military elite class.

But very little 'barbarisation' happened to the Roman populations of either 'Gothic' realm. Or in Frankish Gaul or even in Vandal Africa. Certainly nothing remotely like the evolution of sub Roman Britain to early English Britain. In Continental Europe, the already heavily Romanized Franks, Goths, Vandal, and Burgundian military forces and their commanders who came to permanently control parts of the former Western Empire were only able to make the transition from roving military bands to settled, landed warrior elite by being sufficiently Roman enough for the existing Romans to accept as the new regional power center and then marry, appropriate, and buy their way into the existing landholding Roman elites until the two became one and the same (note: some of the old Roman elites rejected this fusion by going into the Church instead, creating the beginning of some long term Church hierarchies and ecumenical dynasties of their own).