From my understanding from lore posts, judgements and JoJs, when a champion is summoned into the Fields of Justice, it's a union between summoner and champion. The entire reason for the Judgement process that was imposed was to both assess how a particular champion would react to having another entity in its mind and to see what drives said champion to make sure they aren't admitting weak willed individuals (y'know, having to survive the trauma of dying repeatedly over 30 minutes can take a toll on a lesser individual). Champions had control over their actions and were guided by their summoners, but the summoner could also apply enough control to make the champion do certain things. Not exactly how they wanted it done, but enough that if you said "Cho'Gath, go top", Cho could be forced to stroll on up to top as he sees fit to do. I don't think this would normally be exerted, however, as the summoning pact between champion and summoner would require them to have an agreement to be summoned in the first place. How a summoner gained more pacts was through reputation by gaining success and experience through matches, whether sanctioned events, practice or even bets between summoners! The most successful champions were the ones that bonded with their summoners on a much deeper level, unifying their skills to work as a team rather than a puppet and master, which most summoners would already have considering they would have had to form a pact with said champion.
As for Cho'Gath and his supposed power being halved? Let's look at his lore:
"Fortunately, the power of the League's summoning has confined Cho'Gath's presence exclusively to the League of Legends. It is here that summoners use Cho'Gath's Voidborn abilities to help decide the fate of Runeterra."
Cho was trapped by the League, shackled to perform. No where does it state he's at a diminished power or between dimensions or something, only that he's controlled to the point where he won't go ballistic on Runeterra as a whole. His lore as a whole suggests that his power is growing, but not because it's been suppressed. It also suggests that Cho might actually enjoy the League, finding the constant challenge of Runeterra's finest amusing.
But in the end, "Woe betide the day when Cho'Gath grows weary of the League."