Originally Posted by Urobolus
Are they off living their own life? Are they on stand-by? Are they confined for future use?
If you were a champion - a legendary figure that is a hero to some and a villain to others - what would you be doing with your time?
Each champion has their own unique set of circumstances that are independent of the League. Some champions are more closely tied to the League than others. Cho'Gath, for example, doesn't spend much time chilling at the Institute of War; Ryze, in contrast, spends a good chunk of his time at the Institute, honing his magical prowess within the Institute's vast arcane library. Some champions have a very busy schedule that is completely independent of the League. Teemo is quite busy scouting the wilds around Bandle City, keeping it safe from intruders.
TLDR: All champions have lives independent of their time in service to a summoner.
When the call for service to a summoner comes, however, every champion is bound by the following fact: they are ready, willing, and able to assist the summoner who calls upon them in a Field of Justice. The very power of the League - and the fragile tranquility of Runeterra - depends on this.
After this, the champions are whisked back to wherever they were before the summoning.