After you listen to a voice message, it is transferred from our servers and stored locally* on your computer. This means that you won’t be able to listen to it if you sign in to Skype on a different computer.
*Messages are stored locally as .DAT files in the following locations:
Skype for Windows desktop:
C:/Documents and Settings/username/Application Data/SkypeName/voicemail.
Skype for Mac OS X:
Library > Application Support > Skype > Skype_username > Voicemail.