You might consider the "poor man's approach" of Source Control. I've got a free eBook that talks about this at the link below:
Helpful and Handy Links
Just to make sure your questions get nice safe legal answers:
1) Do I just buy another RH program to install? Yes, on your co-worker's PC; make sure he installs it with his login and PC admin privileges. I don't need RH Server, do I? No, you don't.
2) What do I do with my project files? Put them on a mutually accessible server (which we do have)? No. The project source files have to live on the same PC as RoboHelp. When you work on the project, the source files need to be on your PC; when he does the work, they have to be on his PC. Even if you have an "official" copy of the source files in source control, when you check out files to work on, the files are downloaded to your PC.
3) Are there management tips or tricks you might recommend? Rick's method is a good one, especially since there are only two of you. The most important thing is to communicate clearly with one another so that you are always aware of who is editing what.