Do your friend's songs show up in iTunes as "their name's LimeWire tunes"??
If they do, that's just a playlist & the songs haven't been added to iTunes library...when LW's shut down, that playlist doesn't show in iTunes. To see that playlist in your iTunes, just enable iTunes sharing within LW's options.

Is your friend using Vista? Someone using Vista recently said that they had some songs automatically add to iTunes library. How that works I don't know...Vista's different to XP.

If your friend's using a Mac then files can be automatically added to the iTunes library. Mac users might have something within LW's options to choose whether songs are added automatically. Not sure though, I use Windows XP & have to add files to the iTunes library separately.