Remote Desktop allows you to share the clipboard. It even allows you to copy files through a copy and paste mechanism. However, remote assistance does not seem to allow you to do this. Microsoft has made many changes to remote assistance to simplify establishing the connection, however, once connected, I can't even copy and paste text.
Is there any way to fix this?
No, you can't "fix" it. The clipboard redirection feature simply is not available in Remote Assistance for Windows 7.
You will need to use another application to copy/paste data between the clients, eg. an Instant Messenger client