What specifically do I need to do which will enable the appleTV to work on the local network but not have access to the internet?
On the tv, go into Settings->Network->TCP/IP Settings and first note the IP address that is already assigned to your tv.
Then, choose to manually configure the settings. Set the IP address and subnet mask to the same address that had already been automatically assigned to it (that you noted earlier) to make sure that you're in the proper subnet range, and then leave the "router" address as all zeroes.
If you don't know your specific subnet mask, you're probably safe to just use 255.255.255.0, as that's what most home routers use.
This will give the tv an IP address that allows it to communicate with the computers on your own network (since they're all in the same "subnet" and therefore don't need a router between them), but it won't be able to get to the Internet, as it doesn't have an IP address of a router to communicate with.
Question: Does anyone out there in the forums have an apple TV that isn't doing this looping thing at this point? or is it happening to everybody?
Well, I can't speak for anybody else, but the problem has now gone away for me. My tv's have both been on and running just fine for well over a half-hour now.
Since I've changed nothing on my end, that makes it likely that it was something out on the iTunes Store that has now resolved itself (or been resolved).