Short Answer: both.

But firstly that depends on your Definition of Barbarians, if you want to count inhabitants of the Conquered Roman Provinces as Barbarians (which to my understanding would be a yes depending on the Province, as in Gauls would be but Greeks wouldn't) or Roman Subjects. Furthermore it would depend on the timeframe. As im currently working on Mark Antony's Career in the hellenistic East from 41-33 BC I can best speak for the late Republican Era, anything else I'll have to leave up to any colleagues who research those subjects. Given my current Project, I'm gonna go ahead and use examples from his Army.

If you look at Plutarch's Account of Antony's Army going into the Parthian War of 36 BC:

"There were, of the Romans themselves, sixty thousand foot-soldiers,
together with the cavalry classed as Roman, namely, ten thousand
Iberians and Celts; of the other nations there were thirty thousand,
counting alike horsemen and light-armed troops." (Plut. Ant. 37)

As you can see, Troops from the likes of Spain and Gaul were present as far as the Middle East in Roman Armies. Now, if they're differenciated as such by their Origin, it means they dont fight with Roman-Style Armaments, but instead their traditional Equipment they were trained with in their homeland.

Another, maybe better example comes a bit later, when Antony meets with the Median King and they exchange some troops, with the Mede giving Antony some of his Cavalry, trained quite similar to those of the Parthians, which would have been most useful to Antony in another Campaign to Parthia, which unfortunately couldn't happen because of his defeat at Actium and Death shortly after. (Dio. 49.44.2)

As we've established now, there were Troops of non-roman Origin in Antonys Army, and they fought in their original Weaponry. Furthermore we established that Antony commanded many Legions, and those had to be recruited somewhere. Since Octavian held Control over the Italian Peninsula since the Perusine War of 41/40 BC, Antony could not recruit any Legions there (this does not mean there were no Italics under Antonys Command, but very few new recruits from there after the War), but needed Soldiers nonetheless, so he had to recruit Legions in the Provinces he controlled, which were all Provinces east of Shkroda in Albania, as of the Treaty of Brundisium. Any of the new Recruits from those Provinces would be trained as Legionnaires, fight in a Roman System with Roman Equipment.

Concluding, as you can see on the Example of Antony's Army, in the Late Roman Republics Armys there were Barbarian Troops who used their traditional Armaments as well as Barbarians recruited straight into the Legions.

Feel free to put further questions and/or criticisms in the comments

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Literature (mostly in German)

Christoph Schäfer, Kleopatra/Eleanor Goltz Huzar, Mark Antony/Helmut Halfmann, Marcus Antonius/