How to keep screws in a hollow door
I would not mount the latch to the door skin. Even if you can attach it well, chances are you'll pull the skin off the door over time. It's not designed to withstand that sort of stress.
Instead, use long bolts with acorn nuts to sandwich the two sides to the door, or use coupling nuts and bolts from both sides. The latter requires larger holes through the door, but you can use small bolts in the latch plates.
Through-bolt with acorn nut

    |        |
  __|        |__
 |  |        |  |_
(|==|========|==|_)
 |  |        |  |
 |  |        |  |
 |  |        |  |

Screws with coupling nut

    |        |
  __|        |__
 |  | ______ |  |
(|==|=______=|==|)
 |  |        |  |
 |  |        |  |
 |  |        |  |

It's critical that your holes line up, and with four such holes it's challenging.

Clamp or tape the latch plates in place. Verify that they align through the door. Mark the holes and remove the latches.
Drill through the door skin from both sides with a small bit, keeping the drill square to the door face.
Insert the bit through the holes in both faces. Ream the holes to shift position if necessary, aligning the bores.
Gradually increase bit size, finishing with a bit size suitable for your bolts. This will help straighten the final bore.

For what it's worth, my guess is that the door slab was installed backward, with the hinge side used as the latch side. Hollow core doors should still have blocking at the latch area. It's also possible that the slab wasn't intended for this use, and was designed as a bypass closet door, for example. Still, if you install the latch as I suggest it should work fine.