If someone is paying the mortgage on a property but is not on the loan or the title, do they have any legal rights to the property?
If you are the person who isn't on title, isn't on the mortgage and is making payments on the house, the best answer I can give you is " see a lawyer." Beyond that, the answer is, probably not. Unless there is some kind of written agreement with the person who owns the property which grants a security interest in the property in exchange for the monies lent, this would likely be considered an unsecured debt -- if can be proven that there is some kind of debtor-creditor relationship in the first place.
Since you're asking whether the person making the payments has any legal rights in the property, it sounds like that person has made quite a few payments amounting to a large sum of money. That being the case, your question wonderfully illustrates why anyone engages in a transaction like the one you describe and who expects to be repaid should seek the advice of an attorney.