I may be mis-understanding the original question, but I am quite sure that in some jurisdictions, for example, New South Wales, Australia, that the various denominations Parish Registers are formally owned by the local Church parish that creates them, and that access to these registers is not an automatic 'right'. On the other hand, in more than fifty years of interest in entries in the various parish registers, I have not ever been denied access to any original register. In some instances I have located original registers when others thought it had been lost or at least mis-placed.
To me, it is simply a matter of being polite, asking permission and then waiting for the current incumbent to find the time to check and to respond to my request.
I agree with DavidG02's response re matters of record, and I note that Bigamy is a civil crime. I add that our 21st century eyes require precision almost to the nth degree, yet in my own lifetime, people were able to change their names without anyone becoming confused or suspecting the motive for such change. The precision to exact details is currently being associated with identity theft and other adverse impacts on our society, yet in earlier generations the taboo topics included co-habitation v marriage, multiple domestic partners, and fatal self-harm.
Civil registration of Birth, Death and Marriage events commenced at different times across the Australian continent, as in the 19th century there were FIVE separate British Colonies on the mainland, plus TWO further British Colonies nearby (New Zealand and Tasmania).... Each colony established its own civil registration process, however many of the denominations operated in each colony and so the church registers can contain the detail that the various civil registrations often missed. But that does NOT give the civil authorities the right to demand access to the church registers. Of course, the church registers can be handed over to the civil authorities, and this has occurred on many occasions across those seven former colonies, but in co-operation with rather than by demand.
It is quite possible that throughout the British Empire, that for example Church of England parish registers have been accessed by the various civil authorities, and that the processes and protocols associated with that access varies from jurisdiction to jurisdiction and across the decades as well.
I am somewhat amazed at Andrew's point, I am confused by much of his comment. But I can assure Andrew that NSW parish registers commence in 1787, and that they have been accessed by parishioners from their commencement, including for the purpose of what we now know as "family history" but in my forebears time was known as 'genealogy' and 'pedigree charting'.
Cheers, JM.