I'm not aware of any configuration parameters for DFS-R to control the interfaces that it binds to, or to influence how it selects the partner interface to route traffic to. Doing a quick search, I'm coming up with this dirty hack from the Microsoft Storage Team blog (albeit from 2006) that indicates that you should use a HOSTS file on each replication set member to influence their name resoultion such that you effectively "force" them to use the private IP addresses.
This is an ugly hack, and I'm typically violently opposed to using HOSTS files. In this case, though, it may well be the only way to accomplish what you're trying to do.
Rather that doing the HOSTS file hack (which, if you do, you should document so that the next guy who works on it knows why it was done) I have one other idea you might try.
Try putting a host route for the other host on each of the DFS-R replication set computers. If it works, make the the route persistent. I'm about 80/20 in thinking this won't work versus that it will, but it's worth a shot:
Member 1: route add 20.20.0.101 mask 255.255.255.255 192.168.0.100
Member 2: route add 20.20.0.100 mask 255.255.255.255 192.168.0.101
That might just work to get that traffic flowing over the private network. (If I wasn't under orders from The Wife(tm) to get some house work done this morning I'd give it a try myself and tell you if it works... If she catches me writing on Server Fault this morning it will be bad... >smile<)
DFS may replicate back to the source server if the files were changes on one of the other member servers. Remember that even though DFS has a primary server that server is only used for the initial replica set to be copied and distributed to other new member servers. After that yes, files can be replicated back to the host machine.
A good possibility might be any log files or data files that are stored in your App_Data directory could be getting sync'd up to all of the other servers in the replication group.
In general DFS is not the recommended method to deploy updates and application changes to multiple web servers. You might want to have a look into the Microsoft Web Farm Framework which does a much better job managing replication and deployment of application files and other server related settings. It actually does more than DFS is capable of doing in that sense.
Best Answer
I'm using Software Pursuits SureSync to keep a production and standby spare file server hosting roughly 2TB of data in sync at one Customer site. I'm using it in a unidirectional capacity but I know that it supports bidirectional sync (and delta compression, and using the NTFS change journal, and a host of other features). There is an add-on lock manager available that, I would suspect you might also need. I've been very happy with the software (we've been using it for roughly 6 years) and, with a couple of issues that the manufacturer addressed, it has been very stable and reliable.