Best way would be to use something like PHP to get that information and then pass it into Flash. Check the help files though. There may be something in Flash to grab an IP. I just don't know of anything offhand.
yes it wouldn't be too difficult to pass a variable through, but there is a lot of grunt work here, seems someone like flashloaded would have stepped up - there is clearly a lot of demand for user profiling, and many options in the html world. I think before I went to the trouble of coding the routines to record and log all the activity I would split the swfs into separate html windows.