You might be asking about four separate types of modules:
- irreducible Z[G] modules,
- Z-forms of irreducible Q[G] modules,
- indecomposable Z[G] modules, or
- indecomposable Z[G] modules that are finitely generated and free as Z-modules.
I'll assume the last is the main concern.
The irreducible modules of ZS3 are all finite and have an elementary abelian p-group as their additive group. For p=2,3 there are 2 each, and for p>3, there are 3 each.
The irreducible CS3 modules are all realizable over Q. Every such module may be realized over Z, but the two-dimensional representation has two distinct Z-forms, giving four total "irreducible" Z-free ZS3 modules, that is, four total Z-forms of irreducible QS3 modules.
Indecomposable ZS3 modules up to isomorphism are more complicated than the human mind can possibly comprehend. Indeed, even those in which S_3 acts as the identity are much too complex. Luckily they divide up into several types: annihilated by a prime p (then classified by modular representation theory), torsion (more complicated, but basically now p-adic integral reps), Gorenstein projective (Z-free, so covered in the next bullet point), or madness (that is, the rest).
The indecomposable ZS3 modules that are free as Z-modules are classified in:
Lee, Myrna Pike. "Integral representations of dihedral groups of order 2p." Trans. Amer. Math. Soc. 110 (1964) 213–231. MR 156896 doi:10.2307/1993702
There are 10 of them, and the Krull-Schmidt theorem fails for them. Not only are indecomposables not completely reducible, the decomposition of a finitely generated Z-free module into indecomposable summands is not unique. In other words, integral representations of even very small groups are quite complicated.