We report the results of muon-spin spectroscopy (μ+ SR) measurements on the staggered molecular spin chain pym- Cu(NO3)2 (H2O)2 (pym = pyrimidine), a material previously described using sine-Gordon field theory. Zero-field μ+ SR reveals a long range magnetically ordered ground state below a transition temperature TN = 0.23 (1) K. Using longitudinal-field (LF) μ+ SR we investigate the dynamic response in applied magnetic fields 0 < B < 500 mT and find evidence for ballistic spin transport. Our LF μ+ SR measurements on the chiral spin chain Cu (pym) (H2O)4 SiF6 ⋅ H2O instead demonstrate one-dimensional spin diffusion, and the distinct spin transport in these two systems suggests that additional anisotropic interactions play an important role in determining the nature of spin transport in S = 1/2 antiferromagnetic chains.