Is there a way to turn off an external Apple Cinema Display remotely and continue to use the computer? Here's the situation: I often access my Mac Pro remotely using Screen Sharing and sometimes TeamViewer. The problem is that when I do this, everything I'm doing can be seen by those near the physical computer. It also uses a considerable amount unnecessary power to have the displays on but not actually being used.

Is there a way to turn them off and not just blank if possible, while still using Screen Sharing or TeamViewer Free?
I don't think this is possible with Screen Sharing.  (That would go against the "spirit" of the Screen Sharing feature, which is to help people troubleshoot a problem or work collaboratively on something.)

This is not a free alternative, but if you use Apple Remote Desktop ($80), you have the option to "curtain" the screen, which not only hides what you're doing, but locks the remote computer against local access.  It also has many powerful system maintenance/monitoring features.

As far as saving power, could you simply turn off the local display before you leave the location?  You should also set the display to sleep when it's not being used (in the Energy Saver pref pane) -- in that case, it would only be drawing power while you're connected and using it.  Also, if you're using one of the new LED Cinema Displays, keep in mind they are energy efficient devices, so the power draw isn't as much of a concern as with standard (non-LED) displays.