Of course, we can't actually ever know. I'm not expecting wizards to emerge in this thread and provide magical answers. However, given different social perspectives, structures of authority, and structures of governance present during the late Middle Ages, I can't help but think that there must be some kind of academic discourse occurring concerning this matter.
It's my current understanding that people of that time period would consider religious policy much like a modern person would consider domestic social policy -- but that's the problem, see? I have to compare to an existing experience, rather than being able to think "in the time". As I implied initially, I suspect this is at least a little bit necessary, given difficulties that anyone is liable to have trying to divorce themselves from the only experiences they have. All the same, any illumination concerning the topic would be more than appreciated.
This would come under the umbrellas of social and intellectual history. Reconstructing intellectual processes or social milieus is an incredibly difficult thing to do, not impossible but incredibly difficult.
We need ideal sources to do such a thing and in many instances this simply doesn't exist. It also takes an incredible amount of time and effort to put together a study at such a micro-scale. That said it has been done by Carlo Ginzburg in his The Cheese and the Worms: The Cosmos of a Sixteenth-Century Miller. Admittedly it's outside of your period but it is really an exemplar of micro-history in the manner you want to pursue (also might be of use considering your other interests).
The focus of the study is obviously extraordinary (or we wouldn't have records about him) but the manner in which Ginzburg pulls together the sources is masterful and while he might push the evidence too far in terms of identifying some mass agrarian pagan cult the reconstruction of Menocchio's thought process and social milieu is thoroughly marvellous. Here's an interesting blog on the book and it's position in historiography.
As for how practical this is: only when the planets and stars align. It's simply impossible to place ourselves in the shoes of a thirteenth-century priest, a fourteenth-century minor aristocrat, or most women without them producing some abberant record. Margery Kempe is one famous female example from the Later Middle Ages who famously dictated a long and horrible biography to a literate priest before she died (the Booke of Margery Kempe), but Joan of Arc's legacy of historical records (including two detailed trial records) poses more problems than they solve.
Empathy and attempts to immerse oneself in the time are simply not good enough to replace the fact you do think utterly differently from the subjects you study. Your relationship to God (if you have one) is far more open to interpretation than it was in the Late Middle Ages, you have not grown up with the myriad influences which impart implicit knowledge of the society around you. Our records are almost overwhelmingly written, they represent a small sub-strata of a society who weren't producing or preparing their thoughts to be recorded for posterity (and make no mistake people in the Middle Ages were well aware of the concept of legacy and posterity). Good historians can be empathetic, they can make allowances or attempt to get under the skin of their subject, but they must recognise the limitations and dangers of such a process. It takes an admittedly non-scientific process further into the lands of 'art' and ultimately leads to 'fiction'. The reader (of the historian's work) is put even further from the historical record and must then combat the internal narrative and personality of the historian alongside the historical figures they are purporting to emphasise with.
What type of people are you asking about?
Also, GreenReaper, how would you assess the quality of The Cheese and the Worms. I'm intrigued by it, a classmate read it for a review and had...some not so complimentary things to say about the title. It may have been the setting though. Who doesn't love trashing a monograph in a graduate seminar?