A published reference for the claim (right after the question in boldface) is the proof given by Thomason on pages 1657-1658 of "First quadrant spectral sequences in algebraic K-theory via homotopy colimits", Comm. Algebra 10 (1982), no. 15, 1589–1668.  He shows that if the translations are faithful, then S^{-1} S is equivalent to the mapping cone of the diagonal functor S --> S x S, and that this provides a group completion of S.  Baas-Dundas-Richter-R. reviewed this in Lemma 6.1 of our paper on Ring completion of rig categories (Crelle, 2013).