I'm just repeating what others said, but what the hey.
Sounds like you already remapped (since you have the "P3"* stick working, depending how it's connected). Or maybe not since player 2 isn't working (as mame's has no defaults for P2's dual sticks). Because mame's left stick does move the character, and mame's right stick fires, I'll bet it's all about incorrect mapping.
*I find it easier to think the stick as joystick 1-4, NOT player 1-4's joysticks when remapping. So I'd call your layout J3 J1 J2 J4.
I'd say remap general, since the dual stick inputs are different from the normal stick input (IOW, remapping them won't effect non-dual stick games). Start any game, tab, select "Input (general)", select player 1, and scroll down to the p1 right/### and p1 left/### game inputs. The left and right shown prior is the stick, and the "###" is that stick's direction. You want P1 left have J3 mapped, and P1 right to J1. Select each of the eight inputs, and press the corresponding stick and direction in turn. Repeat for player 2 inputs with J2 & J4.
If you do "Inputs (general)" all dual stick games should work. IMO, Cfg Magician is an overkill for this job, but might be the thing if you need to remap some of games differently than other with the same input type.