Unless it adds value for an individual, I'm guessing most folks will not add another form of communication to the ones that they already use that work quite well (email, face-to-face, phones). So, unless I'm mistaken, we have 3 options:
We can promote the value of the Network to our section members, and encourage them to post documents and discussions there. It sounds like some local sections and divisions and governance groups have been successful at making the transition, so perhaps they could share what caused their members to change their behavior.
We can drop the use of email with our members, and mandate that all future documents and discussions will occur on the Network. I'm guessing this is too harsh, and will likely alienate people.
Slowly add more and more of your content to the Network, and encourage people to look there for meeting announcements, minutes, newsletters, etc. In effect, this is a hybrid of the two alternatives above, but would have to be phased in fairly gradually (like maybe over a year or so for groups that have many thousands of members, like the CA Section).