Personal Chronicles. This name gives the idea that your private snap story is like a personal diary, chronicling your unique experiences, thoughts, and feelings that you only want to share with a select few. It has a bit of a nostalgic and personal touch to it.
Well, 'Behind the Scenes' is a nice option. It gives the idea that there are things not usually seen. Another one could be 'Personal Paradise', suggesting a special and private place for you.
Consider the content. If it's mostly about your travels, 'Travelog Snap Secret' would be a good name. You want to pick a name that gives a hint about what's inside the private story.
'My Exclusive Snap Album'. 'Exclusive' makes it clear that only a specific group of people are allowed to view the snaps in this private story. And 'album' gives the impression of a collection of memories or moments that are precious and private.