We study the two-particle dynamics in one-dimensional (1D) electron systems described by the Tomonaga-Luttinger model. By using bosonization methods we compute analytically two different kinds of two-particle response functions. In the first case the creation of the two particles does not involve any other degree of freedom in the system, and the subsequent propagation is described by a standard two-body Green's function. This is relevant to address, e.g, the behavior of recently observed local pairs in 1D optical lattices. In the second case the creation of the two particles occurs coherently with the decay of a third localized particle. Such a condition is crucial to study the Auger transitions in 1D systems like carbon nanotubes, where two holes are left in the valence band after the decay of a core-hole.