This was normal for sidepull brakes back in the day. After you have everything tightened down, you'll need to recenter the brake by turning the center bolt toward the pad that's contacting the rim. There's a couple ways to do this:
1. Use a Park Tool OBW-3 to grip the brake spring and turn the assembly.
2. Put a wrench on the nut on the front and behind the caliper. Turn both nuts simultaneously to turn the center bol.
3. Using a flat-faced punch, tap the spring on one side or the other to recenter the brake.

Have fun and welcome to old-school bike mechanicing.