FitFolks' Secret. This name gives the idea that it's about the private fitness journey of the people in the gym. It has a sense of community within the privacy.
Elite Gym Annex. 'Elite' implies a high - quality, private - like experience, and 'Annex' makes it seem like a special, private part of a larger concept.
Here are a few more ideas. 'My Gym Privacies' makes it clear that it's about the private experiences at the gym. 'Gym Solitude Stories' can be used if you want to convey that these are stories from the quiet or personal moments at the gym. 'Inner Gym Journeys' is also a good one as it gives the sense of the personal growth and experiences that happen within the gym environment, which are private to the individual.
Sweat and Success Tales. It implies that the content of the private story will be about the hard work in the gym and the achievements that come from it, which is very relevant for a gym - related private story.