Here's the LCS rule book:
First of all, technically all pro players do not receive a Riot salary. Rule 2.6 explicitly states that Riot employees are not allowed to play in the LCS. All players are all employed by the respective teams.
Second, there is an explicit provision in section 14.8 for and Academy players (these receive salary from LCS teams) to be able to play in non-riot tournaments so long as they don't interfere with the LCS tournament/league itself. LCS players can participate in 3rd party tournaments too, except that they can't play in the first 3 weeks of LCS in the following LCS split.
Finally, even if Riot completely prevented LCS players from participating in both the LCS and 3rd party tournaments, LCS players aren't slaves. They are free to chose whether they want to play in LCS or not. Similarly, 3rd party tournaments aren't banned from hosting tournaments (like Blizzard pulled with Starcraft/Warcraft in Korea back in the day). The reason there aren't many 3rd party tournaments around is that it doesn't make economic sense to compete with the LCS for team/player participation and viewer eyeballs.