I have no experience of sharing but I've given it some serious thought in the past, so here's my tuppence!
Although it's clear on buying the house what the contributions of the two parties are, the value of the house will change over time, and your contributions towards this change of value could be different.
If both parties were on a mortgage, then when it comes to sell, it wouldn't be fair to look only at the initial investment, the contribution towards the mortgage should also be looked at.
In the case of a mortgagee and a cash buyer, then there are actually three owners here: each of the people and the bank/mortgage provider.
If you were to sell immediately after buying, then the cash owner with 1/3 will own exactly that, but the mortgagee will own only the deposit they put in. The bank owns the rest.
Over time, the mortgagee will pay money. At the point of sale I guess the cash buyer would get 1/3, the bank would get paid off and the mortgagee would get what's left.
Unless - and here's where it could get really complicated - it could be said that one party had contributed more to the property gaining value - by funding an extension or other improvement works, for example. Then that might need to be taken into consideration.
And then if the two parties are married or have kids then there are a whole other set of rules to take account of.
So, not necessarily straightforward!