> the badge will be there, red and proud with a big number on, and you'll see it at every cmd-tab
On KDE, I'd use the "Filters" setting for either the Task Switcher (Alt+Tab thing) or the Task Manager (taskbar), or both. They can show or ignore windows from other virtual desktops, screens, or minimized ones. [Or "activities", but I've never used that feature, and I'm not sure what it's for.]
I usually keep Skype on my second screen, covered with other work, and have the task switcher set to only scroll through not-minimized things on the current screen.
Edit: So "Activities" are something like super-virtual desktops, and appear to be exactly for this purpose, to separate different tasks.