  We demonstrate the universal properties of dissipative Tomonaga-Luttinger
(TL) liquids by calculating correlation functions and performing finite-size
scaling analysis of a non-Hermitian XXZ spin chain as a prototypical model in
one-dimensional open quantum many-body systems. Our analytic calculation is
based on effective field theory with bosonization, finite-size scaling approach
in conformal field theory, and the Bethe-ansatz solution. Our numerical
analysis is based on the density-matrix renormalization group generalized to
non-Hermitian systems (NH-DMRG). We uncover that the model in the massless
regime with weak dissipation belongs to the universality class characterized by
the complex-valued TL parameter, which is related to the complex generalization
of the $c=1$ conformal field theory. As the dissipation strength increases, the
values of the TL parameter obtained by the NH-DMRG begin to deviate from those
obtained by the Bethe-ansatz analysis, indicating that the model becomes
massive for strong dissipation. Our results can be tested with the
two-component Bose-Hubbard system of ultracold atoms subject to two-body loss.
