You would have complete control of the site as long as it's your domain and your own hosting. You can control everything within the social network. You can even charge people to use it.
There are a lot of private people who build the components and work out the errors that deserve the ownership of what they create. They own the components, themes etc. You have their permission to use them as if they were your own, but not to take away their credit for the work. The licence comes with the download.
Just leave the credit to OSSN at the bottom.
I'm relatively new to OSSN, but I can tell you I've been happy with everything so far and the people here are very helpful. It's worth trying out. Best of luck.