The side assignments, Axis or Allied, are determined by the FSO admin CMs. They track which sides your squad has been on for all FSOs and try to see that everyone flys both side, in a fair way. If you are not willing to fly for both Axis and Allied, you should not sign up for FSO. Most squads would like to fly Allied each tour but even the squads which request Axis each tour, will not be granted that request as that's not what FSO is about.
As Gaidin said, ride assignments are determined by the frame CiC with a squads' ride request given consideration, but if you are not willing to take whatever ride is handed out to you, again, you should not sign up for FSO.
The FSO admin goes to great effort to balance the sides based on too many factors to list here, but the concept depends on players willing to take the tougher rides and assignments as well as the preferred ones.
Edit: hehe, see GDs post above