Yes, it comes for free. A short exact sequence of abelian groups, or R-modules (R not necessarily commutative), splits iff it cosplits iff the middle term is the sum of the other two terms (sorry if I misread and you already know this!). The key here is that maps of modules and be added/subtracted:
Let the middle maps in 0→A→B→C→0 be f:A→B and g:B→C. Then if there's a splitting q:C→B, then (1B-qg):B→B is a projection onto A as a submodule of B, i.e. a cosplitting. Together the splitting and cosplitting exhibit B as the direct sum of A and C.
A dual trick shows that a cosplit sequences are split: if p:B→A is a cosplitting, then (1B-fp):B→B is a projection onto a submodule of itself which is isomorphic to C via g, i.e. a cosplitting, so again B is the sum of A and C via these maps.
More generally, this works in any abelian category. One way to recognize this instantly is via Freyd's Exact Embedding theorem, which roughly implies that you can pretend a diagram in an abelian category is a diagram of R-modules for some R.
... whoah... are you guys related?