Yes, it comes for free. A short exact sequence of abelian groups, or modules, 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!):
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. A dual trick shows that a cosplic sequences are split.