Stream input is controlled by callbacks in each mission. To keep a mission from posting information to the Jive activity stream, clear the Send Callback checkbox.
Disabling the callback checkbox for a mission will keep Gamification from sending new information about mission completion to the Jive activity streams, including the user's Inbox as well as Social News. The user will still see the point totals and badges for the mission reflected in the Reputation tab.
To enable and disable posting to the activity stream: