How exactly are you launching the program? Do you get any output when launching it from the terminal?
anthy is just the IM backend. Which input method framework are you using? ibus, scim, scim-bridge, uim?
skype appears to use qt4, so do you have all the necessary qt extensions for your framework installed also? ibus-qt4/scim-qtimm/scim-bridge-client-qt4/uim-qt? (I'm going from my debian sources here, *buntu may have different package names.)
What are your IM environment settings? What output do you get with the command
I've never used skype, and now thanks to Microsoft I never will, but some googling shows me that it's probably an environment setting issue. This page, for example: http://code.google.com/p/ibus/issues/detail?id=613