Anyone can make their own email public by adding a thread to Streak. You cannot make someone else's email public by adding your version to Streak.
If you see the 'Request access' notification in a thread shared to Streak, additional messages exist on the thread that do not exist in the inbox of the person who originally shared.
After your access is granted, you will be able to view, but not respond to, the additional message(s).
An email sharing example
In this example below, Andrew has sent an email to one person on the same Streak team (Nikola) and one person not on the Streak team (Frank). Andrew then added his version of the thread to Streak. The external contact (Frank) then replies all and also replies directly only to Nikola (and not to Andrew, who added the thread to Streak):
Because the message between Frank and Nikola is not in Andrew's inbox, Andrew can see the additional message exists - but cannot view the contents. Andrew can make his own email public by adding to Streak, but he cannot make Nikola's email public.
Looking at Nikola's perspective, we see the feedback from Streak that not all messages are shared:
Andrew can 'Request access' and the banner will update:
... and Nikola will receive an email notification:
... and Nikola shares, Andrew receives a matching notification:
Finally, Andrew can see a 'view only' (cannot forward or reply) version of the message that was shared: