This is kind of one of the big questions in provincial and religious history of the Roman Empire. Since you mentioned Romans and Germans as your example (and since that is what I am most familiar with), I'll run with that. That means I will focus on the Northwestern provinces of the Empire (modern France, Germany, Austria, Switzerland and Britain mainly) to find an answer to this. It's not easy to answer straightforwardly, since we have no written sources by, say, a German that tells us that he considered a Roman deity as the same as the equivalent Germanic one.

Most people familiar with classical antiquity will probably at least have heard of the term interpretatio Romana. If they're not familiar with the term, they will - as you mentioned - be able to equate the Roman Pantheon with the Greek one - Iuppiter-Zeus; Iuno-Hera; Minerva-Athena; Apollo-Apollon; Mercurius-Hermes and so on, as codified by the Roman poet Ennius in the late 3rd-2nd century BC. The Romans, as most other mediterranean civilizations, were well able to introduce foreign Gods into their own pantheon. Diana, for example, was an italic Goddess venerated in the Alban Hills before she was introduced to Rome, we can see this happen throughout Roman history with gods and godesses such as Kybele, Isis, Mithras and so on. In large part this was because taking a foreign cities tutelary deity could be part of the conquest - the deity was invited to leave her original city, thus stripping the city of its protection, in return for veneration at Rome.

The Roman pantheon was not a system that was terribly strict. Deities were polyvalent, they had different aspects and forms in which they could be venerated because those aspects were responsible for different things. Iuppiter, for example, in imperial times was most often venerated as Iuppiter Optimus Maximus, Iuppiter the Best and Greatest, the highest state god; but he had other aspects or forms, such as Iuppiter Fulgur, the lightning bolt, or Iuppiter Stator, the Iuppiter that stays the flight (of the army) among many others; later of importance Iuppiter Dolichenus, popular among soldiers and tied to the city of Doliche where his cult originated. Most deities still had specific or primary functions that characterized them, but they appear as flexible deities that are open to new forms of cult that appear (and disappear) over time and stress different aspects of the respective deity (or bring new ones into them).

I don't want to discuss this at length, but the main thing to take away here is that deities were a flexible thing - they may have had primary functions, but their concrete form in which they could be venerated by a specific individual at a specific time was highly circumstantial. Specific forms of a deity could belong to specific cities, or areas, or social groups.

So with this in mind, I want to talk about the Roman northwestern provinces, which saw the contact between an already by that time fairly established Roman pantheon and a plethora of local deities. I will use the term 'Celtic/Germanic' here, but keep in mind that this does not mean that there was a firmly established, monolithic 'celtic' or 'germanic' culture or pantheon that the Romans met on their conquest, but it's still useful as an umbrella term here I think. The gods venerated in these areas were polycentric and polyvalent, they appeared in different areas and with different aspects, some were highly localized, just to keep that in mind.

In fact, the very origin of the term interpretatio Romana, which conceptually fits what you are asking about, comes from Tacitus talking about a cult practiced in the territory of the Nahanarvals (eastern Germany):

In the country of the latter is a grove, consecrated to religious rites of great antiquity. A priest presides over them, dressed in woman's apparel; but the gods worshipped there are said, according to the Roman interpretation, to be Castor and Pollux. Their attributes are the same; their name, Alcis. No images, indeed, or vestiges of foreign superstition, appear in their worship; but they are revered under the character of young men and brothers. [Tac. Germania 43]

Here, deities that are revered under the name of Alcis are identified as the Roman deities Castor and Pollux, the divine twins. Their attributes are the same, as Tacitus says, they are young men and brothers, so they must be Castor and Pollux. By the late republic, as a result of the conquests and the growing importance of control of the religious (since it was so deeply entwined with the political sphere), Roman religion had aquired a kind of a claim to universality - this was a way to overcome certain historic contingencies, the different and heterogenous cults that made up the Roman pantheon, its more or less random composition, the different local cults and so on. By laying claim to a universal pantheon, these contingencies could be rationalized and overcome. Identifying other Gods with Roman ones secured their existence across wider space and ensured their survival in different contexts while at the same time widening the power and influence of the Roman forms of worship. If the Germans were really just worshipping Castor and Pollux by a different name, they could be more easily integrated, and it heightened the perceived legitimacy of the Roman gods - after all, even the Barbarians knew about them, and worshipped them.

The early Christian author Minucius Felix commented on this:

Thus we see that in all empires, provinces and cities the people have peculiar religious customs, and their inhabitants venerate peculiar fos, such as the Eleusines Ceres, the Phrygians Magna Mater, the Epidaurians Aesculap, the Chaldaeans Bel, the Syrians Astarte, the Taurians Diana, the Gauls Mercurius, and the Romans all Gods (universa Romanos). [Min. Fel. 6.1]

In short, it meant that all the Gods that were venerated in the provinces under new Roman control, were just Roman gods by another name, from the perspective of the Romans - this is what meant by interpretatio Romana. Caesar writes about the the Gauls that their favourite God was Mercurius, and after these Apollo, Mars, Iuppiter and Minerva, but he gives no Celtic names for them, and he doesn't need to, since to the audience he writes to this is not relevant - no matter their actual customs, traditions or mythology (that we don't know), they are just the Roman gods worshipped by other people under a different name.

This is of course a very imperialistic way to go about the whole business. For the Romans this was for the most part not that big a problem. But what about the new Romans, those Celts and Germans living in the conquered territories? We don't have terribly much evidence surviving from the actual practice of provincial religion, but one big category are votive altars and inscribed votive gifts. They show us that the provincial population mostly venerated the traditional Roman gods, most popularly in the Germanic provinces, to prove Caesar true, Mercurius, but also the Capitoline trias (Iuppiter, Iuno, Minerva) and other popular Roman gods such as Apollo, Hercules and so on. Often, they appear with just their Roman names. The provincial population seems to have accepted the Roman interpretation, worshipping gods by their Roman names, in the appropriate Roman fashion.

Names give power ower things - if you name something in a certain way, you prescribe how that thing is viewed. This is also important in the context of Gods - I already mentioned that this Roman way to go about identifying foreign gods with Roman ones and giving them their Roman names is a very imperialistic form to handle the whole business. It ignores the viewpoints of the conquered and presses their deities into a pre-formed mould. Webster [p. 157] said that 'interpretatio remains a game played by imperialist rules', and I tend to agree with her. It mainly informs us about how a latin-literate, non-indigenous elite viewed the Celtic or Germanic gods, and many of the equivalences made come from a purely functionalistic point of view: god of sky and thunder = Iuppiter. It's the view of the imperial elite that all gods everywhere are the same as the own. Mythological and religious differences tend to disappear and become invisible to us, the indigenous belief system disappears under the Roman overlay.

However, luckily, we have many instances were the Gods are not only named according to Roman fashion. Sometimes, they are paired with Celtic or Germanic gods - a popular pair was Mercurius and the celtic goddess Rosmerta, whose function is not precisely clear but who often appears together with Mercurius in northeastern Gaul. Often though, and this is where it gets interesting, we find names added to the Roman gods. One such example would be Iuppiter Taranucnus. Taranucnus was another celtic deity associated with thunder and lightning, consequently identified with Iuppiter. We don't know the inner world of the dedicant, we don't know the mythology behind Iuppiter Taranucnus, or his precise attributes, or what the dedicant really believed in - her inner world is closed to our view. But clearly whoever erected the altar to Iuppiter Taranucnus (CIL III, 2804) - I'm being needlessly obscure, we know at least her name, Arria Successa, a proper, Roman, name - didn't find it enough to inscribe the name Iuppiter alone. She associated him with the Celtic deity Taranucnus, and if you choose to view this in the context of the discourse about power (over the name of Gods) in the provinces you can see it as an act of resistance against an overbearing purely Roman belief system. That may be a bit too extreme, though at the very least that Iuppiter Taranucnus was different from other Iuppiters (otherwise you wouldn't have to add the epithet), and this difference was connected with the Celtic deity named Taranucnus.

[cont'd below]