We present high-resolution Magellan/MIKE spectroscopy of 42 red giant stars in seven stellar streams confirmed by the Southern Stellar Stream Spectroscopic Survey (S 5): ATLAS, Aliqa Uma, Chenab, Elqui, Indus, Jhelum, and Phoenix. Abundances of 30 elements have been derived from over 10,000 individual line measurements or upper limits using photometric stellar parameters and a standard LTE analysis. This is currently the most extensive set of element abundances for stars in stellar streams. Three streams (ATLAS, Aliqa Uma, and Phoenix) are disrupted metal-poor globular clusters, although only weak evidence is seen for the light-element anticorrelations commonly observed in globular clusters. Four streams (Chenab, Elqui, Indus, and Jhelum) are disrupted dwarf galaxies, and their stars display abundance signatures that suggest progenitors with stellar masses ranging from 106 to 107 M o. Extensive description is provided for the analysis methods, including the derivation of a new method for including the effect of stellar parameter correlations on each star's abundance and uncertainty. This paper includes data gathered with the 6.5 m Magellan Telescopes located at Las Campanas Observatory, Chile.