I don’t have “private” Challenges yet, but it is my intention to eventually allow people to create their own Challenges and also choose to have those personal Challenges be public or private.
This is a new addition to the idea of Challenges, and I haven’t given it any thought yet. I expect that the initial release of personal Challenges will allow you to provide a title and description, with no “rules” for what can be included in the description. I don’t know if I’ll learn that I need rules - we’ll all find that out together. 
Yeah, I expect I’ll include the ability to allow others to join your Challenge. Originally, I figured that was the public/private selection - but now I’m seeing that you could want a Challenge to be public (people can see that you are doing this thing) but you would not want to allow people to join (look at me to this thing vs look at us do this thing).
There’s also another request that I’ve received, for collective Challenges. The idea there is that a group of people are trying to complete a city together so if PersonA completes StreetA and PersonB completes StreetB then the entire group has collectively completed both StreetA & StreetB.
There’s no app, yet: Create a CityStrides app