Only user have permission Edit can choose the value.
In the Issue Screen, Jira have two mode ( Issue Create, Issue View).
In Issue Create mode, permission Edit is required. The user can choose the values in the custom field.
In Issue View mode, the user have permission view or edit will see the value of the custom field.
Click on this custom field to go Edit mode.
If user don’t have permission
Field Security Group Picker get the value from the chosen custom field on Issue. The application will process the value.