I purchased and installed the Greenshot from the App Store. It installed successfully and I can see the "G" icon on the status bar. However, which I either click the icon and choose any of the screenshot options or attempt the same using any of the shortcut keys, *nothing happens". No screenshot is taken, no Destination Picker (the default) is shown. I have no idea why it isn't working. I've also used Greenshot on Windows going back many years, so I'm familiar with the basic features and functionality (I understand Windows and Mac versions are not 1-to-1 on functionality at this time).
For what it's worth, the Preferences (menu) and Quit options do seem to work. In the case of the former, I can't take a screenshot to verify it's really working, but I can get to the options and change them.
Macbook Pro 2016
Mac Sierra 10.12.3