I can't speak for other areas of the Roman Empire, but in post-Empire Iberia, after the fall of the Gothic kings in 711, and with the massive wave of Moors who conquered, there was a relatively 'tolerant' position taken by the Moors and Caliphate in Cordoba. Ibero-Romance peoples were able to keep their culture, religion, and customs at will...however, if they wanted to climb the social ladder, they would be wise to morph into Islamic ways. In fact, linguistically there's a very rich area of 'mozarabes'--those Ibero-Romance peoples who took on Arabic/Moorish customs, names, and even language...but remained Christian. Their language appears in jarchas--Mozarabe poetry, often of love, rebellion or saddness, which is written in Middle Arabic or Hebrew, save for the last 2 stanzas. These last 2 stanzas are written in Arabic or Hebrew script, but are actually Ibero-Romance. The Mozarabes were bilingual and bicultural, by all discussions that I know.
Now, this is all taken from two linguistics sources: Rafael LaPesa's Historia de la lengua espa