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In a house with shared ownership, if one person moves out and the other assumes mortgage, how do we determine who owns what share in the end?
The ownership of the house depends on what the original deed transferring title at the time of purchase says and how this ownership is listed in government records where the title transfer deed is registered. Hopefully the  two records are consistent.  In legal systems that descended from British common law (including the US), the two most common forms of ownership are tenancy in common meaning that, unless otherwise specified in the title deed, each of the owners has an equal share in the entire property, and can sell or bequeath his/her share without requiring the approval  of the others, and joint tenancy with right of survivorship meaning  that all owners have equal share, and if one owner  dies, the survivors form a new JTWROS. Spouses generally own property,  especially the home, in a special kind of JTWROS called tenancy by the entirety. On the other hand, the rule is that unless explicitly specified otherwise, tenancy in common with equal shares is how the owners hold the property. Other countries may have different default assumptions, and/or have multiple other forms of ownership (see e.g. here for the intricate rules applicable in India). Mortgages are a different issue. Most mortgages state that the mortgagees are jointly and severally liable for the mortgage  payments meaning that the mortgage holder does not care who makes the payment but only that the mortgage payment is made in full. If one owner refuses to pay his share, the others cannot send in their shares of the mortgage payment due and tell the bank to sue the recalcitrant co-owner for his share of the payment: everybody is liable (and can be sued) for the unpaid amount, and if the bank forecloses, everybody's share in the property is seized, not just the share owned by the recalcitrant person.  It is, of course, possible to for different co-owners to have separate mortgages for their individual shares, but the legalities (including questions such as whose lien is primary and whose secondary) are complicated. With regard to who paid what over the years of ownership, it does not matter as far as the ownership is concerned. If it is a tenancy in common with equal shares, the fact that the various owners paid the bills (mortgage payments, property taxes, repairs and maintenance) in unequal amounts does not  change the ownership of the property unless a new deed is recorded with the new percentages.  Now, the co-owners may decide among themselves as a matter of fairness that any money realized from a sale of the property should be divided up in accordance with the proportion that each contributed during the ownership, but that is a different issue. If I were a buyer of property titled as tenancy in common, I (or the bank who is lending me money to make the purchase) would issue separate checks to each co-seller in proportion to the percentages listed on the deed of ownership, and let them worry about whether they should transfer money among themselves to make it equitable. (Careful here! Gift taxes might  well be due if large sums of money change hands).