Another option could be 'Intimate Narratives'. This name implies that the stories are something private and close to you, and it has a bit of a mysterious and alluring feel to it.
For a travel journal private story, 'My Solo Travel Escapades' could be a great name. It clearly indicates that it's about your personal travel experiences.
Other possible names are 'Personal Parable', 'Secret Script', 'Intrinsic Story'. 'Personal Parable' can be used if the private story has some sort of moral or lesson within it that is personal to the teller. 'Secret Script' makes it seem like there is a written or unwritten script of a story that is secret. 'Intrinsic Story' gives the feeling that the story is an essential and private part of someone's life or experience.
Some funny names might be 'The Private Prattle' or 'My Confidential Chronicles'. Well, when choosing a name for your private story, it should reflect the essence of what the story is about, in a more private and personal way. These names give a sense of secrecy and exclusivity, which is what a private story is all about. You can also be creative and combine words that have a special meaning to you, like 'My Personal Puzzle' if your story is full of mysterious elements.