"The grid of the flyswatter is electrically charged to a voltage of between 500 and 1500 V DC," according to Mr. Wiki. The regular ones are not fully rectified though, so they should produce pulsed DC current. If they are producing HV AC then they would require ground connections, as do the big, stationary ones with UV lights you hang near your window.
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I'd bet AC. There's no particular reason to rectify it.
If it's high voltage AC then it would need a ground connection, like the stationary ones with UV lights.
I have one with a dc boost converter that charges a capacitor. it's DC. You hear it charging up with a <whrrrrrrr> sound as the voltage gets higher. I have another that just outputs a high frequency sound, and I'm guessing its a dc>pulsed dc or ac upconverter with no capacitor storage. It just whines at the same frequency no matter what the load on the circuit. If they weren't hundreds of kilometers away I'd measure them for you... The big ones that hang with a light inside them just run medium current high voltage ac through the grill.