Persons who share bodies still have distinct legal and social existences, so if one commits a crime, the other is entitled to walk free while awake as long as they come back before sleeping - but how do they prove it?

Is this a realistic cultural adaptation? In most human societies if you are stuck working or living with someone your social existence is somewhat shared. A person from your clan doing something bad is a also bad for your own reputation. If someone from your family committed a crime some legal traditions would hold you responsible. It seems much more plausible that society would consider the two people living in the same body to be legally treated at least like a married couple or brothers where in some past ones.

Given your constraints and assuming no cheap and easy test of distinguishing them, of all historical examples I can think of, only modern Western culture with its hyper-individual liberalism would bother with the impracticality of treating the two people like two fully distinct individuals. And even then they would have to give a family-like if not legal guardian-like relationship for the issue of making medical decisions. Not sharing your place of residence and ownership over it would be impractical, though perhaps there would be a strong norm of not going into the other guys part of the house.

Also as a minor note the culture would probably develop a norm of some sort of marker (perhaps clothes, jewellery) or face paint to show which of the two persons in currently in control. The distinction would be more or less universal not simply individualized so even strangers could tell this was two different persons. Think more "Ah I see your patron god is the first twin Jahu. Your cohabitor was here yesterday." instead of "Aha James always wears his leather jacket you must be Harry!". Using the wrong marker would probably at least as taboo as cross-dressing was in some past cultures.