Lets say you flip a coin for initiative in Skirmish.

Is the player who wins the toss obligated to start with initiative, or does he then get the choice of who gets initiative, much like X-Wing?

According to the rules, you are supposed to determine who gets initiative randomly. Not randomly choose a person who gets to pick initiative.

The suggested method is to roll a blue die and the highest accuracy gets initiative. If you tie, roll again.

See: https://community.fantasyflightgames.com/topic/176753-question-on-initiative/