For a build blog this year, our team will be using Google Blogger, which is pretty powerful, user friendly, and free. Your domain would be something like “mydomain.blogspot.com”, but for ~$5-$10/yr, you can purchase another domain and have your blog linked to it.
Alternatively, if you already have a website and host, you can create a submain and have it direct to blogger (like blog.yourdomain.com)
Either way, Google would essentially be running the service for you, which takes a bit of headache out of it. I am fairly certain there are controls to limit access as well if you choose to go that route.
As far as privacy, we will opt to leave ours public and be pretty liberal about posting details on our progress. In a 6 week build season, I think it is unlikely that your success will hinge on a competitor’s ability to copy your design… but instead on your ability to execute your own design. Additionally, we feel a good blog is a great place to direct school administrators, parents, sponsors, etc… to allow them to follow our team through the season. If you limit access, you’ll either have to find a way to grant these folks access, or miss out on a good PR opportunity.
On the Wiki side, I have run MediaWiki before, and it isn’t too difficult (though, more involved than setting up a blog). You will need to have access to a machine to act as a webserver, either by purchasing web hosting, or by setting up your own machine as a webserver. Again, I would point you to a COTS solution like using Google Documents, which is a great way to allow multiple people to share information, edit files together in real time, and control permissions levels if needed.
Feel free to PM me if you have any questions.
-Steven