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Recording other people working out at the gym is rude, super lame, and creepy. Though, people who want their privacy may be dismayed to learn it may not be illegal.
Since most gyms are accessible to the public, it is not a violation of privacy laws to record someone in a gym's common area. Typically the law does not prohibit watching or photographing activities that are outside a person's reasonable expectation of privacy.
Taking a photo of someone on the elliptical, a reasonably public view, would likely be legal. A photo of them in the locker room ... not so much.
As long as the subject is a clothed gymgoer in a common public area, the law provides little protection from secret Snapchats.
State and federal laws are mostly silent on recording people at the gym. But each gym has its own policies regarding photography and video.
Here are some recording policies at gyms which have locations nationwide:
#Gymselfies are extremely common, so it may be hard for each gym to enforce these rules. Many people also record themselves to check their form or add a fitness video to social media. You could get in trouble for filming this at the gym. To be polite to others, at the very least you should try to make sure no one else is in the scene or ask their permission before you record.
If you catch someone recording you with their smartphone (or vice versa), it will be up to the individual gym to deal with the problem.
Breaking these rules is likely considered a breach of the gym membership agreement. So the worst that could really happen is that the gym could cancel someone's membership and kick them out.
Bottom line: Recording at the gym is weird, but may be legal in a public, clothed setting.