To get two instances of Skype working, I run the second one via su:
$ sudo su another-user
$ /Applications/Skype.app/Contents/MacOS/Skype
This gets a second Skype to work easily: it correctly saves the history in the account of another-user, etc — no clashes whatsoever.
Unfortunately the pasteboard does not work in that su'd instance of Skype. It's not like it's using a separate pasteboard, seems like it cannot use a pasteboard at all, since copying and pasting into the same Skype does not work either. I suspect it is some kind of permissions problem.
I wonder if anyone has an idea of what is wrong and how to fix it.
(To be clear: this is not specific to Skype. If I run TextEdit.app instead of Skype in a su'd session, pasteboard does not work just as well.)