Some cool names might be 'Exclusive Escapades'. It gives the feeling that the things in this private story are like special adventures only for you or a select few. Another option could be 'Intimate Insights'. This name implies that the content of the private story is something very personal and full of deep understanding. And 'Solemn Soliloquy' is also a good one. It makes it seem like the private story is a place where you can have your own quiet, personal monologues.
Friendship's Hidden Hideout. This name shows that the private story is a hidden place where the secrets and special moments of your friendship are stored, away from the rest of the world.
Secret Scribbles. This name gives the idea of a private place where you can write down your thoughts and experiences that are just for you or a select few. It has a sense of secrecy and intimacy.
You could call it 'Secret Chronicles', which gives a sense of mystery and privacy. Another option is 'Personal Tales', which is simple yet effective in indicating that these are your own private stories. Also, 'Hidden Narratives' is a cool name as it implies that the stories are not for public view but are hidden away.