For a long time, we believed that, in some form or another, that there was a process of conscious and guided Romanization - that is, a process that led to the conquered people (with the admitted exception of hellenic cultures) becoming, in effect Roman. They would adopt the Latin language and Roman customs, Roman dress, Roman forms of municipal and tribal administration, jurisdiction, luxuries and so on, which superseded their own culture.

This traditional teleological view of Romanization as the rightful and inevitable triumph of a superior culture over lesser ones, who only appeared as passive recipients, is no longer commonly accepted. Rather, when we look at the individual areas, we see a highly heterogenous picture of cultures interacting with each other in ways that are impossible to generalize over the whole Empire, so it really is impossible to answer your question with a confident yes or no. My answer would be a hesitating, conditional yes. When new territories became part of the Roman Empire, what Rome was primarily concerned with was:

• how can we organize tax collection

• how can we provide external and internal security for the newly conquered territories and the Empire in general

• who are the local leaders, and how can we make them do what we want

This meant, that for a large part of the population, not much would change immediately. Their leaders might have to learn Latin to communicate with their new masters (or continue speaking Greek), they might serve in the Roman auxiliaries and become citizens after 25 years, new settlers or traders might come to their lands, their gods might be called by new names (some by their old and new at the same time), their gods might be prayed to by the Romans now also, and so on. Other changes might be more drastic: laws might change, depending on the forms of local organization, land might be confiscated for new colonies, taxes/tributes might be very high. On the other hand, there were also increased opportunities, such as an unprecedented spatial mobility - a Syrian auxiliary might find himself settling down in Britannia or Germania after his service. Traders could rely on Roman infrastructure in a very large space. However, local culture might continue for quite some time. There is evidence for 'code-switching' behaviour, where members of the leading caste might adopt Roman manners, dress and so on when they were dealing with Roman officials, but switch back to tribal dress, language and culture in their own civitas.

An example would be Pannonia, which contained a mix of 'celtic', dacian and illyrian ethnicities. It had been commonly assumed that Pannonia had been thoroughly romanized by the end of the 1st century. From there, we have an astonishingly high number of statues on tombstones, which display the deceased, presumably as he or his descendants wanted to be viewed by others. Along with celtic names, we find a large number of women in tribal dresses, wearing ethnic accessoires such as certain hats, broches and rings, while the men often display themselves as togati, toga-wearing Roman citizens - some even alongside their wife wearing ethnic accessoires. On the other hand, they quite rapidly accepted the Roman burial customs, which were different from the earlier ones, and the inscription formulae alongside the Latin language (at least on the inscriptions). In some cases you can see the changes along generational lines, where a father with a celtic name would be buried by his son Marcus.

It is important in this regard not to think of Roman culture of something that was opposed to local culture. Becoming Roman did not mean no longer being a Boian, or Dacian, or Batavian or whatever.

In other areas, the picture might be different. For example, in Hispania and some central Italian regions, the local language and local inscription formulae remained in use long into the time of the Empire. If you are interested specifically in Gaul, a seminal work in that respect would be Greg Woolfs 1998 Becoming Roman, in which the whole idea of Romanization gets criticized quite a bit as well (in fact this is one of the most important works in the field still).

There were changes that affected the whole Empire, and the change to Empire itself largely affected the culture of all Roman territories. The constitutio antoniniana, which gave all inhabitants of the Empire Roman citizenship, was one Empire-wide event that wasn't without influence on culture.

There is currently a lively debate going on about how to frame the process that went on when new territories became part of the Roman Empire, with a lot of concepts being thrown around. A good, if by now a bit dated, summary is Alföldy, G. (2005): Die Romanisation - Grundbegriff oder Fehlgriff? Überlegungen zum gegenwärtigen Stand der Erforschung von Integrationsprozessen im Römischen Weltreich*, in: Visy, Z. (ed.): Limes XIX. Proceedings of the XIXth international Congress of Roman Frontier Studies held in Pécs, Hungary, September 2003. Pécs, p. 25-56. I personally think that Romanization is still useful as an analytical category, however we have to widen our field of view when looking at the processes that occured in the contact between Rome and newly conquered/controlled territories. Perhaps it is also helpful not to view it judgmental as 'others becoming Roman', but rather as the socialization of people into a multiethnic, multinational Empire with a "rather astonishing cohesive capacity" (Peachin 2011, The Oxford Handbook of Social Relations in the Roman World, p. 12)