Skype uses resources from your PC when its idle to no point in even keeping it on unless you are using it.
But yeah, some of this crap is starting to look scary.
If I'm understanding this "Eavesdropping attack" correctly the only thing scary about this is the sensationalized media reporting it. Use a voice canceling MICROPHONE and turn the volume just high enough so only YOUR voice is heard, then there is no way for skype(or anything else) to drop eaves on you.
I remember years ago before LCDs became mainstream and there were similar concerns of being able to reproduce the image on a CRT monitor by the EM radiation it gave off and it was suggested to put the CRT monitor in a Faraday cage of course the fact that in order to do that the person dropping eaves had to be sitting next to you or in the cube directly next to yours.