What to do if the display is scrambled during Screen Sharing in iChat

I ran into and resolved an issue with iChat/Messages screen sharing that I'm posting here in case anyone else runs into the same problem and needs help.

PROBLEM: When I started screen sharing through iChat, my display of the remote screen was scrambled:

• The top inch of my display had random streaks and noise
• The menubar and desktop of the remote screen appeared below that inch
• Clicking on an icon didn't work or selected the wrong thing. On further examination, my mouse pointer was offset by the noise at the top of the display, i.e., if I moved my pointer to the upper left corner of the noise, I could select the Apple menu, but if I clicked on the image of the Apple menu I could see, it wouldn't work (or would select some other icon).

CAUSE: The person I was screen sharing with had just bought a second, HD display. In Display Preferences, they had it set to Mirror their original MacBook display. But they wanted to fill the larger display with the same desktop, and set it to a different screen resolution than the MacBook's 1280 x 800.

SOLUTIONS:

• Have them go to System Preferences/Displays. Resolution menus will appear on both screens. The second "mirror" display should be set to the same resolution as their main screen. Quit iChat, reopen it, and start the screen sharing again.
• Uncheck "mirror displays". Quit iChat, reopen it, and start the screen sharing again. Now, you will see both displays, so they will be very small on your screen...this may be useful for certain things.
• Have them shut down their Mac, unplug the second screen, and restart, so the second display isn't an issue.