There is the work De Dea Syria which has several such stories, such as that of Stratonice, but it is a Greek book, not a Roman one.
In Roman history, there are various one-liners describing divine interventions, but these generally are public in nature and do not concern the beliefs of private persons.
In Roman literature, little record remains of accounts of divine revelations with the exception of the Metamorphoses of Apuleis, in which it is important theme that especially dominates book XI. In that chapter, the hero, Lucius, begins describing his revelation as follows:
Necdum satis coniveram et ecce pelago medio venerandos diis etiam vultus attollens emergit divina facies; ac dehinc paulatim toto corpore perlucidum simulacrum excusso pelago ante me constitisse visum est. Eius mirandam speciem ad vos etiam referre conitar, si tamen mihi disserendi tribuerit facultatem paupertas oris humani, vel ipsum numen eius dapsilem copiam elocutilis facundiae sumministraverit.
(I had hardly closed my eyes when suddenly from the midst of the sea a divine face emerged, displaying a countenance worthy of adoration even by the gods. Slowly it appeared until its whole body came into view and, the brine shaken off, a radiant vision stood before me. I shall try to describe its marvellous appearance for you too, if only the poverty of human speech allows me the means of expression, or the deity herself supplies me with rich abundance of rhetorical skill.)
Trans: J. Arthur Hanson
In that book can be found considerable material describing the effects of divine revelation, apostasy and repentance. Some of these ideas are expressed through the sermons and sayings of the priest and others through Lucius himself.
For example, Lucius says things like "Quis venerabili continentia rite servatis" and "Ergo quod solum potest sine piaculo ad profanorum intelligentias enuntiari referam" referring to his atonement, and things like "divinos tuos vultus numenque sanctissimum intra pectoris mei secreta conditum perpetuo custodiens imaginabor" (I will store your divine contenance in the secret places of my heart forever.)
You have to remember that the Romans valued action, not thought. The Judeo-Christian mentality is to focus on a person's attitudes and presumed psychological viewpoints (are they "repentent" for example). The Romans did not think like that, and focused on a person's behavior. Did they perform the right sacrifices and so forth. To a Roman, piety was expressed by behavior, not by what the penitent was saying about their state of mind.