I can confirm the same problem. The error message for Android screenshots goes something like:
Couldn't capture screenshot.
Taking screenshots isn't allowed by the app or your organization
I had to exit PowerApps to at least take a screenshot of the error message.
It doesn't seem to depend on the specific PowerApp. In fact there doesn't seem to be any PowerApp (advanced) setting regarding screenshot. Even taking screenshots of PowerApps main screen/PowerApp listing does not work. The PowerApps App itself still doesn't seem to have any setting beyond "Optimize images for upload".
My colleage told me he could recently do a screencasted Reflector presentation from his iPhone so the problem seems to be Android specific.
This is totally a major issue and unacceptable! If you introduce features/s**t like this, you at least have to add a setting to switch it on/off.
Last tested: Android 8.1.0 on Nexus 6P PowerApps 3.19093.17 Session ID 69e4e2cb-0253-4b1b-bf4e-23864a9ff285