It's purely random now, unless you set the starting player in the tables option. I think it would be bad, if the "official" rule was implemented without giving a rating bonus to second player. In that case, people who care about their rating wouldn't want to rematch after a win and rather find the next (often same) opponent via matchmaking.
Back in the isotropic days I was always a bit annoyed by this rule for exactly the reasons Markus is describing here. Iso was a little more advanced, but people could still influence their 1P/2P rating by playing on or not.
I also thought it could have a nice solution: instead of looking at the previous game this session (what you do IRL) you could look at the previous result between this exact set of players. Obviously IRL that would be a nightmare because of bookkeeping, but online there is this database.
Anyway, I haven't discussed it with Donald yet, unless of course I am doing that right now.