Hi,
The Apple Remote Desktop client is installed on the Mac anyway.
The Bit that does the controlling is the bit that costs.
However the Apple Remote Desktop engine is the bit that makes the Screen Sharing work and is used by iChat to it's Screen Sharing.
NOTE.
Apple Remote Desktop is enabled in Sharing as the remote Management item
This cannot be enable at the same time as the Screen Sharing item is enabled.
To Link in Screen Sharing you need to go to ther Finder's Go Menu and enter an IP.
This IP should be the Public IP of your sister's Modem/router.
That Router should be set to pass data that come of the port for this VNC connection to the Mac she is running.
This would normally be done by setting up Port Forwarding in her routing device, as BobHarris has already said
iChat's Screen Sharing adds a VNC connection to display the Shared Screen alongside an Audio Chat and negotiates it way through firewall and routers (if you can Video or Audio Chat already).
iChat uses a Random port for this and it pays to use UPnP in the routing device at both ends to get it work
(you can't Port Forward a random port and iChat requires 19 other ports to be set up).
Screen Sharing does need to be Enabled in the Video menu of iChat.
10:19 PM Saturday; January 14, 2012
Please, if posting Logs, do not post any Log info after the line "Binary Images for iChat"
iMac 2.5Ghz 5i 2011 (Lion 10.7.2)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
"Limit the Logs to the Bits above Binary Images." No, Seriously