Disclaimer: I'm not a historian, but i do take an active interest in Mithraism and mod r/Mithras.

The cult of Mithras had a (mostly) centralised belief (as much as we can infer from inscriptions/motifs) and a decentralised structure, though its presence was primarily in Rome. For example, while gradings are often talked about in relation to Mithraism, we don't actually know if all Mithraea were uniform in their implementation of them. Manfred Clauss, one of the foremost experts on Mithraism, writes in The Roman cult of Mithras, p.131:

"Although it is not always clearly stated, most previous accounts assume that all Mithraists were members of one grade or another. ... Should we not rather conclude that in the cult there were, on the one hand, the great majority of Mithraists, who were simply initiated once, and, on the other, a small group of holders of the different grades, whom it would be appropriate to speak of as 'priests'?"

In addition to our questions of gradings, we likewise have questions of the uniformity of beliefs and practices. Certainly, the tauroctony is mostly uniform, but in some areas of the empire, we see alterations. The same is true for other scenes like Mithras' birth (e.g. at a site near Hadrian's Wall, Mithras was equated with Phanes of the Orphic tradition, though it should be noted this comparison is also true for one inscription in Rome and in a list by Zenobius).

Enough on the uniformity of structure and belief though, what about practice? Mithraea were built to represent caves - specifically the one in which Mithras sacrificed the bull. These would often be constructed near sources of water, and this was common in and outside of Rome. We again visit Clauss, who states in The Roman cult of Mithras, p.73:

"...the importance of water for all manner of ritual purposes is revealed by the water-basins and cisterns, by the representations of Oceanus, and also by the evident desire to locate temples in the vicinity of a river or a spring. Water-basins were clearly part of the basic equipment of all mithraea."

Mithraea, in Rome (and Italy), were often built in public spaces as opposed to private dwellings (like many examples we have in Europe). That said, it still retained an element of secrecy, and even writings about the cult at the time were prone to speculation.

Mithraism never achieved official recognition by the Roman Empire, though it was accepted. Later in its history, powerful individuals did support Mithras in an attempt to introduce a pagan revival (like Emperor Julian and Senator Rufius Caeionius Sabinus), though this does not necessarily translate to actual membership of the cult.

So what was Mithraism in Rome? It was a decentralised group of Mithras-followers who primarily worshiped in secret, though at temples mostly dedicated in public spaces like bath houses. They may or may not have differed to other Mithraea throughout the empire in gradings and beliefs. As a cult, they didn't enjoy state recognition, but they were accepted until their late decline, after which they experienced a small revival before perishing completely.