Many cities big enough to have a Health Dept (e.g., New York, but many smaller cities, too), sometimes offer testing through their clinics. In a big city, there are many small clinics so you can go out of your neighborhood. Smaller cities, I don't know. I guess it could be embarrassing to go to the clinic and see your neighbor at the receptionist's desk:
"Hi, what are you doing here?"
"Uh, ... wrong door ... I was aiming for the MiniMart next door to buy a magazine."
"How did you get up here to the third floor?"
"Uh, I thought, maybe, their high class magazines were on an upper level."
Of course, you might take a different approach:
"Hi, what are you doing here?"
"I've been bangin' your daughter and I'm a little concerned about her rash so I'm here for a test.