when you buy 'your' half then you decide for which club the player is going to play that season and how long the co-ownership deal lasts (1 year or more, also depends on length of players contract). Player signs two distinct contracts (possibly with different wages) with the clubs. At the end of the year you can ask to delay tre decision (if contract is long enough) or go on with the silent auction. You can also buy the other half during the season and players is all yours! A 'third' team needs to buy both halves in separate deals to buy the player.
usually it's a good way to get players at lower price, while is also an investment who will leave you with some value (different from a loan). If you are a big club it's a nice way to 'control' the market esp for young players (often big clubs would co-own young players and let them develop at lesser clubs before buying the other half)
It gets trickier though... You can also sell your half to other teams, or include your half as an asset in other deals (same as exchanging players). IRL... it's even trickier as it is often a way to inflate values of players and generate profit from nothing, cant do this in FM though... I think!
...just to understand ...