Some good names would be 'Gym Reveries'. It gives the idea of personal thoughts and daydreams that might happen at the gym, which is very private. 'Exclusive Gym Tales' is another option. The word 'exclusive' implies that it's only for a select few or just for oneself, which is what a private story is about. Then there's 'Gym Privacies'. It directly mentions the privacy aspect and is a short and catchy name for a gym private story.
Well, there are quite a few options. For example, 'Fitness Secrecy'. This name suggests that the story is about something private related to fitness, which is perfect for a gym private story. 'Gym Confidential' is also a great choice. It has a mysterious and private connotation, making it suitable for a private story about the gym. And 'Personal Gym Chronicles' is a name that makes it seem like a collection of personal stories from the gym.