We have no Mithraic writings, either scriptures or histories, which makes this a difficult question to answer (it's part of being a mystery cult). Most of the information we have on Mithraism is archeological, not manuscript, which is usually the wrong sort of evidence for this sort of question.

Being a mystery cult, members would most likely approach those they saw as worthy and invite them to become initiates. As for who they invited, and how Mithraism spread, the traditional theory has been that much of the spread of Mithraism was due to its popularity among the Roman military, and while it's still a valid theory there isn't an overwhelming amount of evidence to support it.

The movement of soldiers across the empire would explain the movement of Mithraism, but it isn't the only possible explanation. The Roman Empire functioned in large part by being highly dependent upon social networks, both within cities and across trade routes, and Mithraism could have simply spread through the networks of patronage, family connections, trade and such that helped shape Roman society.