Depends on the period in history. In Roman times coins were not as they are today. It was not a design, but a weight. Anyone could make their own coins, and put whatever they wanted on them, but the weight was what determined their value. So in the case of the Romans conquering some neighboring nation, they would just take their money, and the fact that it had another guy or god on it didn't really matter.

Sorry, we don't allow throughout history questions. These tend to produce threads which are collections of trivia, not the in-depth discussions about a particular topic we're looking for. If you have a specific question about a historical event or period or person, please feel free to re-compose your question and submit it again. Alternatively, questions of this type can be directed to more appropriate subreddits, such as /r/history or /r/askhistory.

Maybe you could choose a specific conquest to focus on? As it stands, it's literally unanswerable.