By default, the clipboards of the host and client computers will be synchronized.
To toggle between synchronized and not synchronized, when in a Remote Control session, click the Customize Toolbar button on the black remote control ribbon. The Customize Toolbar button looks like an arrow pointing downward. Go to Connection Options and select Sync Clipboard. You will see another icon appear in your Customize Toolbar area. Clicking that icon, which looks like a clipboard, will toggle between a synchronized clipboard and an unsynchronized clipboard.