Co-ownership is a bit like like a stock market futures contract combined with a loan. What's important to remember is that co-ownership deals are usually finite arrangements. There is a term put in place when the deal ends and both teams must "go to the envelopes" in terms of buying out full stakes in the player. In other words, if Roma co-own a player from Bari, then a few years from now, Roma and Bari will submit secret bids for the other half of the player. The highest bidder pays that amount to the loser and then takes control of the contract. So Roma can choose to outbid Bari for the player because they value him highly or even not bid at all if they do not want to take on the contract (i.e. the player has not lived up to expectations). In the event of tied bids, the club who currently has the player on its roster wins.
Both clubs can usually also buyout the other's stake at any point in the deal, but this is like a normal transfer where bids can go back and forth.
In the case of a player already contracted to a smaller club, co-ownership is usually pursued by a bigger club to buy a long-term stake in the player. The player will usually continue to play with the small club and develop as a key player in that side. The bigger club gains the advantage of getting a long-term developmental prospect with the option to fully purchase him during the co-ownership or when the deal ends. In most other leagues, teams buy a player and then loan him back (like Manchester United did last year with Wilifried Zaha to Crystal Palace). In Italy, the co-ownership process achieves the same ends but for a longer term and allows the big club to hedge its risk slightly better.
In some cases, a player on a big club will be sold in part to a smaller club as a way of not giving up on a player. Rather than multiple loans, the player is allowed to settle and develop on a smaller side in the hopes of being a better asset later in the deal. For example, Inter signed Adriano from Brazil and then loaned him to Fiorentina when he was behind strikers like Christian Vieri (IIRC) on the starting squad. With the loan expired, Inter and Parma then agreed on a co-ownership wherein Adriano thrived for two years in a yellow shirt. The end result was a win-win for both clubs as Parma received great service from Adriano for two seasons and then received Inter's buyout for full rights. Inter had a player with known discipline issues thrive in an environment where he found peak form and were able to take him back when the side needed another striker.
Obviously, not all co-ownerships work out well, just like other transfers. But the takeaway really is that co-ownership is like a long-term development agreement where both parties agree to share the benefits or costs of a player future value.