You can only assign tasks and appointments. Events do not have the ability to assign a team member. All Calendar items can have participants. This is not the same as a responsible party.
To assign a task or appointment to an individual on the team, choose their name for the available team members when creating the task or appointment. The responsible party will be notified via in app message and email that they have been assigned a calendar item. They will either need to accept or decline the invitation.