Yep, this just went live yesterday, I believe:
https://support.google.com/stadia/answer/10001744
I get the impression that sharing functionality hasn't been added yet. The Help Center describes how to view or delete captures but nothing about sharing them:
https://support.google.com/stadia/answer/9582306
You might consider using the Send Feedback option in the app to let the team know how eager you are for this capability to be added.
@Blaster : Share still isn't available, but it is now at least possible to download clips and screenshots via Chrome.
Seems to be working now. At least I discovered the share button in the app TODAY.
Yep, this just went live yesterday, I believe:
https://support.google.com/stadia/answer/10001744