Before going to examples, here are some general comments:
a) Proving that a finitely generated group is Hopfian is usually pretty hard unless the group is residually finite e.g. finitely generated subgroups of $GL(n,\mathbb R)$ are residually finite, hence Hopfian by an old result of Mal'cev.
b) A common method in proving that a group $G$ is co-Hopfian is to use an invariant of $G$ that is multiplicative under passing to finite index subgroups. If $G$ has a nonzero such invariant, then $G$ has no finite index subgroups isomorphic to itself. For example, if $G$ is the fundamental group of a finite aspherical CW-complex of nonzero Euler characteristic, then $G$ has no finite index subgroups isomorphic to itself.
c) If $G$ is the fundamental group of a closed aspherical manifold, then $G$ has no infinite index subgroups isomorphic to $G$ (look at top-dimensional homology).
d) Euler characteristic, signature, $L^2$-Betti numbers, simplicial volume are are multiplicative under finite covers of closed aspherical manifolds, so if $G$ is the fundamental group of a closed aspherical manifold with say nonzero signature, then $G$ is co-Hopfian.
Here are some specific examples to add to Richard's example of one-ended torsion free hyperbolic groups. All of the following groups are linear, hence residually finite, hence Hopfian.
The fundamental groups of closed locally symmetric spaces of nonpositive curvature without local flat factors are co-Hopfian because they have nonzero simplicial volume thanks to a result of Lafont-Schmidt.
If memory serves me, it is possible to figure out which geometric 3-manifold groups are co-Hopfian. For example, the $SL_2(\mathbb R)$-Seifert fibered spaces have a certain invariant detecting volume of the base $2$-orbifold which is multiplicative under finite covers. Check papers of Pierre Derbez in arXiv.
Fundamental groups of some nilmanifolds are co-Hopfian, see my paper here.
Delzant-Potyagalo classified co-Hopfian Kleinian groups (in real hyperbolic space of any dimension). See here.