If they are in a team, they get a paycheck from the organization. The organization gets money from sponsors and investors. Some organizations take a percentage of tournament winnings, some leave it all to the players. It really depends on the organization.
Stream sponsorships are more often with full-time streamers rather than esports players who have sponsors from the organization. If an esports player streams and gets donations, I think all of that money goes to him, but it could also depend on the organization.