Yes and no. The Romans did have a single standardized currency that maintained a pretty impressive stability. However, "big data" archaeological examination of coin and particularly the project of matching coins to their mints has shown that coins weren't really regularly issued. Coins would be issued and used to meet the needs of the administration, say to pay the soldiers or for construction projects. It wasn't as standardized as today.

Another caveat is that Roman coinage didn't always drive out local coinage. This is a complicated issue that I'm not terribly familiar with, but Egypt maintained its own monetary system and several cities in the east issued their own coins (albeit to Roman weight standards). I have not really seen a good account of the economic implications of this.

This is true until the late 3rd century, at least. After that, the coinage, though already fairly standardized in the precious metals, was even more thoroughly unified. In fact, just what you ask did happen - the Sardes excavation reports a London mint follis of Constantine I.

Edit: this was supposed to be a reply to /u/Tiako