What is displayed on a stream is entirely based off the Asset you configure within the stream at the assets of your activity. For example "profile.USER_ID" will show all profile activity of USER_ID. If you want to show all global activity only then set the Asset for the stream to "global" then ensure it's configured to display global activity (should be the case by default).

You can't filter global posts. They ignore asset filtering and are always displayed on every stream. So you won't be able to filter a stream to a specific users global posts. In short it is all or nothing. Your regular users shouldn't be able to make global posts to begin with. Global posts are a moderator feature and your regular users should not have moderator permissions. Instead you should use 2 different streams for what you're wanting.

One stream is their normal profile Activity tab. Another stream is an activity field or activity page with an asset of "profile.[user_id].ad". Now you'll be able to do what you're wanting. Those posts will show like normal on their profile, but can also be filtered down specifically using "profile.USER_ID.ad" for example.