There is no ticket income from Private League games. Supporters have no additional opportunities to earn income than non-supporters.
PL games also must be played at "normal" effort, so they do not impact on the team's enthusiasm either. The only "advantages" one can get from PL games is the opportunity to test out tactics - but how useful that may be is also somewhat impacted by whether or not the level of competition in your PL is similar to the level of competition in your regular league.