I have learnt how to solve it from this article:
http://brennan.offwhite.net/blog/2007/01/18/fixing-copypaste-for-remote-desktop-sessions/ (This link has died, so I've updated it to point to an archive.org snapshot. -ed.)
To make it short:
You have to stop the application called rdpclip.exe in your remote machine. Then you go to Run, and write rdpclip.exe to restart this app. The link between the clipboards is recovered.
The clipboard implements what is called " clipboard viewer chain " and is application-controlled, that is to say, that every application that is added to this chain is responsible of passing on notification messages to the next viewer in the chain, there are apps that don't pass the messages and this causes the problem (in Vista is solved).
(there is a wider explanation here: