Sometime very recently Google made the process of sharing your Stadia captures much easier.
As noted by 9To5Google, it's now possible to share your Stadia captures without having to first download them. For nearly a year, this has been the only way of sharing your Stadia captures. Go the website or app, find the capture you want, and save that image or video. Then proceed to share it anywhere you like.
While that process was better than having no ability at all to share your content, it wasn't ideal. Google has finally fixed this. And even if it is late in the game, it's here now.
Sharing your Stadia captures can be done from the app or website
No matter where you're playing Stadia, it's possible to capture your gameplay. When it comes to sharing, you can do that almost anywhere.
While you can't share your content from the UI while playing on Chromecast, you can share pics and videos from both the Stadia app and on the Stadia website. This also works from both the Android and iOS apps.
To share something, just go the captures section in either of these locations and tap on an image. There should now be a share button in the bottom left corner, and by tapping that it creates a sharable link.
If you're doing this on Android, after the link is created you should see a popup box that contains the link, which you can copy, and another button to share that link using Android's native share menu.
At this point just pick where or who with you're sharing the picture or video clip and continue as you would for sharing anything else on the platform. Obvious if you're sharing on the web, just copy the link and paste it somewhere.
You can also make captures private at any point in time
If you ever change your mind about a shared image or video and want to keep it to yourself, you don't have to delete the content.
After you make an image or clip sharable, the popup box has a toggle that's labeled as "share by link." If you toggle that off, this doesn't give you another way to share the content as the description implies.
What it actually does is make that content private so only you can see it. Even if you've shared the link with someone previously. It's a quick and easy way to keep your content to yourself at any point. And you can just as easily make it shareable in the future by tapping the toggle again.