This is likely not a problem with skype as much as it is with your Operating System. Depending on the settings of a computer, an application can get total control of the running sound device, and with such control, said application can mute everything but itself, among other functions that can cause any other applications to have trouble playing audio. Cubase (a professional audio editing software), for example, mutes every single application except for itself when it is running, for some reason. This IS changeable in this software though, and if this is what is happening with your skype, it should be changeable there too.
Somewhere in your audio/sound settings, there is probably a radio button that says something like "Allow Applications to take exclusive control of audio device" or the likes of that. If it is checked, un-check it, and skype should not then interfere with the use of other audio-playing apps.