I kinda do this all the damn time. I have two Twitter accounts, a public one and a private one. So do a lot of my friends. It's not uncommon to find myself replying to a tweet from a friend's public account with the private one, because I want to say something not meant for the entire world to hear.
These private accounts all started out for talking about our sexytimes, but tend to very quickly lose focus and be for just 'talking with friends instead of the world'. And occasionally sharing half-naked selfies and whatnot.
This is a real pain in the ass if you only use Twitter via their web interface, but usually insanely easy if you use an app.