They talk to each other and agree amicably what the fair split is, taking care not to sour any future relationship they might want to have.
This is obviously the "sensible approach", but as you say - it's a bit more complicated than that.
The present and future relationship of the parties may be the paramount concern (of one or both). If it's not, then all sorts of factors come into play other than "fairness".
Is the other party happy with the present half share arrangement? If yes, then the offer to buy them out will have to be too good to refuse (i.e. more, maybe a lot more, than they paid). I presume that they can't be made to sell.
Can a suitable replacement be acquired for the same amount? If the alternative for a happy party is to acquire their own item, then this may be the price (£435) they demand to exit the arrangement. The alternative for the unhappy party is to go and buy their own item for £435. How much premium can the happy party negotiate for their half share plus the unfettered enjoyment of the other half?
Is the item special or unique? Then all bets are off. Until the happy party wants to sell, the price is whatever they demand.
This is why partnership/shareholder/family business agreements are so important (and so much "fun" when things go wrong).
If they can agree that a sale should happen and the sticking point is the price, then there may be solution. Many agreements include a "shotgun clause". The party (A) wishing to buy out the other party (B) nominates the price. B must either sell at that price or buy A's share at that price. This is like "You cut, I'll choose".