They're state-registered nonprofits that can write donations off their taxes and have the mayor and town council give speeches at their parties and donate to their coffers. When Corona hit, these clubs (Rifle Club, Football Club, Church Ladies Club, etc.) all got called to the mayor's office and they organized purchasing groceries for shut-ins and stuff.
They're like an auxiliary arm of the local government, rather than something 💯 private or 💯 public. They also influence the government, as the government is dependent upon them for society to function well. German government relies heavily on this sort of volunteerism.
