Different jurisdictions, especially different countries, will have different rules about this sort of thing. I'm in Canada. Here, ownership of documents is often determined just like any other piece of property. Disputes come up if there are arguments about how someone acquired the document and whether it was a legitimate acquisition, but they are a property. So if I find my great-grandfather's will in the provincial archive, I can see it but I can't request that they give it to me because that paper belongs to them.

Often, when documents are given to historical societies or archives, they are given with a deed of gift or some similar document that formally transfers ownership from the donor to the donee. For example, I work for the Anglican Church of Canada. When a parish has documentation of a certain age/quantity/nature, it is to be transferred to their diocesan archive. Even though it is institutional policy that this is the standard practice with these sorts of documents, when the documents are transferred, the parish still completes a deed of gift for the documents to the diocesan archive. This means that if someone questions how the archive came into ownership of those documents, they can produce a form that shows the date and under whose authority the parish gave those documents to the archive.

If the donation was being made from a private citizen or private organization to a historical society or other archive, I am sure that there would be similar documentation showing the chain of custody and ownership of those documents. I am sure that if you requested your ancestors' documents from the historical society, they would tell you that the documents are their property and that you are welcome to peruse, view, and otherwise make use of them, but ownership and custody of the documents will stay with the society.

Source: Spend a lot of time as a custodian of archival material and with professional archivists in my work.