The leader selects the mission from all the available missions on the team. All of the group can enter and do the mission. It doesn't matter who clicks the glowies or completes each element, but you must keep going until you get to the mission complete notification and the blue exit button. If you do not get these then the mission is not yet completed.
After completion, if another team member has the same mission from the same contact then they are given the option to complete the mission from their list as well (or not).
I've always found it works this way as the default without altering any menu options.