2 prong outlet with 3 wires
I used a continuity tester and found that 


The right wire #1 comes from the switch. The switch's other wire is the hot, so when the switch is turned on, the power flows through the switch, then to this wire that feeds this outlet on that right wire.
The bottom-left wire #3 comes from the other outlet, and it's a neutral in that outlet, so basically it is the continuation of that outlet's neutral. (I'm not sure why they hooked it up this way rather than using a cap to tie it together with a pigtail to the outlet? But I guess electrically it's the same. I think I will rewire it the way I'm used to seeing.)
The top-left wire #2 must be the neutral.


As far as why there's voltage on #3, I'm baffled! I don't know how hacky this is but I used my jumper cables to hook to the grounding rod outside so I had a definite ground (this is a 2 prong outlet so I didn't know what else to use as ground), then tested the wires and #1 had 120V as expected with the switch on, but #3 had 70V even when it was disconnected from this outlet! I'm going to have to call an electrician to investigate this further because I have no clue why.