By private do you mean an internal network using private addressing:
10.x.x.x
172.16.x.x - 172.31.x.x
192.168.x.x

As with any host on an internal network you just have to insure that the host can communicate with a SMTP server, whether it is on the internal network or an external / public network. That is just simply a matter of network routing.

EDIT:
An easy way to determine if your internal hosts can communicate with an SMTP server is to use telnet to connect to the SMTP server. From a Windows command prompt on the same subnet as your Cx array type the following:

This tells telnet to connect to the server on port 25
c:\ telnet 25

You should see a response similar to the one below. this is the SMTP server beginning the conversation.
220 smtphost.domain.com ESMTP Mon, 13 Apr 2009 09:31:33 -0700

Then type the following to say hello to the SMTP server:
helo

If you got this far then you can communicate with SMTP server from the local private network. See:
http://support.microsoft.com/kb/323350/en-us

this is what i did. I have a similar setup where we keep all of our EMC gear on private network (10.x.x.x). My management hosts has two network cards, one on private network and one on public. On my management host i am running IIS, specifically the SMTP module. I've setup SMTP relay where my EMC gear that uses email, will email my management host which turns around and forwards that email to my company's SMTP server. I've secured port 25 where it only excepts requests from 10.x network.