Ahhh -- I think I finally figured it out. (At least by doing what I have below, it's working!)
I'm using a combination of two plugins -- ThreeWP Broadcast along with Role Scoper.
I made sure the particular user making the post to broadcast was listed as a user on both of the sites in the network (with a Subscriber role).
On each individual site, in the "Roles" section (from Role Scoper), I made the particular user a Post Editor for the particular category where she'd be posting (on both sites), and made sure that category was named the same on both sites. (I also made her a Post Editor as the default role for all new posts.)
And then when she goes to add a new post on one site, she has to select:
1. The appropriate category in the "Categories" section
2. The "Broadcast taxonomies also" checkbox in the "Broadcast" section
3. The other site to post to in the "Broadcast to:" section.