Which side of these (absolutly terrible) calipers is the leading edge? I'm being stubborn about thowing this bike together with zero money.
Not sure what you mean by leading edge here. Those are calipers for a pocket bike, e.g. 50cc motorcycle. If this is for a pocket bike, you might have better luck on a motorcycle forum. That said, each caliper can only fit in one direction when the rotor and wheel are installed. The actuation arm dictates which is front and rear (or left and right for dual front rotors) by where the cable is coming from on the bike. Provided those calipers aren't seized and the actuation arms are pictured in the neutral position, the cable must pull the arm away from its neutral position, i.e. counter-clockwise and clockwise respectively (L-R, as pictured).
Maybe you know this, but just in case: you're missing a cable stop on the left caliper and you'll need cable-fixing bolts. Also, I get that you're trying not to spend any money, but check the brake pads before installation. Even if the pads have a bunch of material left, looking at those calipers, my guess is that the pads are contaminated or worse glazed. You will never get the squeal out of contaminated pads.