If someone has a birth certificate and a marriage certificate and someone else wants it. How is it decided who owns this document

Who is the other person? What country? Usually nobody has a right to the original document but the person who's name is on it. Excepting in cases of minor children where the Parent or legal guardian has it as well.

For a Marriage certificate after a divorce the judge decides how retains the original and who gets a copy.

But in both of those cases the "Birth Certificate" isn't the real birth Certificate that is held by the government who ownes it... what most people think of as a birth certifcate is actually a Certificate of Birth that has less information and is subordinate to the master the government has. Same situation with a marriage certificate... the Government has an original as well, and what's theirs is theirs.

I have my grandmother's baptismal certificate (she was born at home, so no birth certificate) and marriage certificate. She's been dead since 1958, so I guess I "own" them.

If my husband dies, I "own" our marriage certificate.

It's not a question of ownership, but a question of who has possession of it. All birth, marriage, and death certificates are to be filed at the county office, so maybe the county "owns" them.

Why are you saying "own"?

It's a trivial question; the value is the price paid to obtain an official copy ($10-$20 maybe?), so in most cases it's not likely to get to court. But in theory, it would be the person who paid for it who owns it. Then he or she either gives it to someone else (who then becomes the owner), or dies and his/her heirs or devisees "own" it.

So, to answer the original question, apply the above rules to trace the provenence (history of ownership) to determine who owns it.

Like other items of personal property, it would be common that someone not necessarily entilted to it simply took it at some point (for example when the previous owner died). Despite the saying that "possession is nine tenths of the law" (This saying is really a presumption: if one possesses the thing, that person is presumed to have acquired it through either gift, purchase, or inheritance. This presumption can be overcome by sufficient evidence.), strictly speaking one cannot establish ownership by simply taking/stealing something.

It's similar to how ownership of paintings, antiques, etc. is determined.