The activity tab is a unique usage. It displays your activity, activity posted on your stream, your connections activity, and the activity of things you're following only when viewing the stream as the profile owner. If you are not the profile owner it only shows that profiles activity and not their connections activity or their followed activity. This is the same behavior you'd see on Facebook home page vs profile page. It does this automatically, but only if Asset has not been set in the activity tab. Below is a breakdown of what asset it's using.
Viewing Self: user.connections,following
Viewing Other: user
The only way to mimic this behavior with other streams is to use IF substitution conditions in your asset since it does support substitutions. Example as follows.
[cb:if user_id="myid"]user.connections,following[cb:else]user[/cb:else][/cb:if]
The condition checks if the user id being used for the substitutions is the viewing user. See the below substitution tutorial for further usage information.
www.joomlapolis.com/documentation/279-community-builder/tutorials/18353-using-substitutions-throughout-cb