Some good ones could be 'Confidential Chuckles', 'Exclusive Escapades' and 'Restricted Ridiculousness'. These names give a sense of secrecy and fun at the same time. They can make your private stories seem more interesting and unique. For example, 'Confidential Chuckles' implies that there are some really funny and private things that only a selected few can see, which is exactly what a snap private story is all about.
Private Moments Gallery. This name gives the idea that your snap story is a collection of personal and private moments that are precious to you. It has a bit of an artistic and exclusive feel to it.