Two siblings own a home with mortgage. What is fair for both parties? Should the sibling in the home pay the entire mortgage including HOA, property taxes, and insurance? Or should there a be a % split by the siblings so one sibling in the home pays x % and the other sibling not living in the home pays y %?
The siblings need to wear two hats: An owners hat and a tenant hat. Both siblings have the owners hat, and it sounds like just one of them will have the tenant hat. One of the siblings will be only an owner, the other will be both and owner and a tenant.
The owners need to "charge rent" to the "tenant" which should be based on a normal, fair, market rent for the house. Just like they would if they were renting to anyone else. The "tenant" pays rent to the "owners," who share this equally, along with all the usual landlord/owner expenses that go along with it. The owners should share HOA, property taxes, insurance, repairs, etc. The rent amount should be revisited periodically to ensure that it remains at market. No family discounts :-)
The "tenant" will pay for normal tenant things, including rent, utilities, general upkeep, yard work, etc.
They should keep a separate set of books as landlords, receiving income from the tenant, paying expenses, etc. If there is "profit" left over, then they can share that - though I would recommend putting it into an account for repairs down the road. If there is a loss, meaning the rent doesn't cover all the expenses, then both owners decide together on what to do, and contribute equally to the expense.
The other option is the non-resident sibling imagines themselves a bank providing a mortgage (perhaps an interest-only one?) to the resident sibling for the amount of equity they have in the house.
The resident sibling then pays the amount of interest expected on a mortgage of that size - which will of course be much less than the rent arising from the other answer. The resident sibling then pays all the taxes, repairs, bills and fees - and if the house gains or loses value, they take the entire gain or loss.
The benefit of this is if the resident sibling wants to get improvements done to the property: If they want to get a luxury kitchen fitted they can save up and do so at their own expense - instead of having to split the cost with the non-resident sibling, who'd be justifiably reluctant to spend $$$$ on granite countertops they'd never get to use.
The disadvantage is the two siblings are subject to very different risks: Even if today the siblings find it fair the resident takes any gain or loss in the house's value, they might feel differently in 10 years time, if one of them has been made radically richer or poorer by the deal. If this option appeals to you, why not just have the resident sibling remortgage to buy out the non-resident sibling?
It is very complicated and depends on circumstances. If one sibling is not interested in the property, then they can arrange the other one to pay it and own it including paying half of currently payed load to the one leaving the property ± market valuation changes.
Or they can both pay the load and share ownership but then follow any country laws of sharing a property.
Or they can give property under rent and have the tenant sibling go somewhere else.
Or see other answers for other possible arrangements.
Many options. In no event it makes sense though for tenant sibling to pay loan and then both siblings own it.