I can only speak to League, as I do watch and enjoy Dota 2 but am not heavily invested in the scene.
Riot pays each team a total salary. The minimum a team pays each player is $12,500 a split, which is half a year. While this is low, players do not pay for housing, transportation to the matches, food (but do pay for things like going out to eat on their own, otherwise they are provided food), or game equipment. The team is the one that provides all that, and Riot pays them a certain amount to cover those costs, as well as pays for things like a coach and manager. Note that that $12,500 is the minimum and is the amount kicked in by Riot only. Some teams also pay their players a salary on top of that, and I believe Riot also pays more to higher caliber teams. So while some lower tier teams are paid the minimum, many players make six figures a year. This does not include prize money, but prize money pales in comparison to Dota 2. Also many stream, and a few of the top tier players can pull in another six figures just from streaming.
The LCS, or League of Legends Championship Series, is the league all these players play in. This, at least in 2013, was not profitable for Riot. Riot views the LCS as a way to grow eSports and their game. It's advertising for League of Legends. While it may be profitable now or in the future, it was not the goal of Riot to make bank off of the eSports scene. Sure, they'd love to, and it may be a goal, but it is/was not THE goal.