We classify all possible JSJ decompositions of one-ended torsion-free two-generator word-hyperbolic groups. One of the corollaries is that such groups have virtually cyclic outer automorphism group. We also show that a torsion-free one-ended two-generator hyperbolic group has a strong version of "strong accessibility", that is any chain of splittings over cyclic subgroups terminates and there's a global bound on the length of such chains.
We construct an example of a two-generator one-ended subgroup of a torsion-free word hyperbolic group that is not co-Hopf.