I am struggling with this issue too.
From Apple support:
"Can photos and videos be saved or downloaded from a shared photo stream?
Yes. After photos are shared, anyone who can view the shared stream can download and keep them permanently. Videos can only be downloaded to a Mac or PC. On an iOS device, videos are streamed from iCloud and can't be downloaded."
WHY would this be the case. It's MY content...MY video that I shot with my iPhone. I saved it to the cloud so I could get it on my iPad. It is there via the cloud. But what if I want to show it to someone when I don't have WiFi? Why can't I download it to "My Videos"?