  We take the paradigm of interacting spin chains, the Heisenberg
spin-$\frac{1}{2}$ XXZ model, as a reference system and consider interacting
models that are related to it by Jordan-Wigner transformations and restrictions
to sub-chains. An example is the fermionic analogue of the gapless XXZ
Hamiltonian, which, in a continuum scaling limit, is described by the massless
Thirring model. We work out the R\'enyi-$\alpha$ entropies of disjoint blocks
in the ground state and extract the universal scaling functions describing the
R\'enyi-$\alpha$ tripartite information in the limit of infinite lengths. We
consider also the von Neumann entropy, but only in the limit of large distance.
We show how to use the entropies of spin blocks to unveil the spin structures
of the underlying massless Thirring model. Finally, we speculate about the
tripartite information after global quenches and conjecture its asymptotic
behaviour in the limit of infinite time and small quench. The resulting
conjecture for the ``residual tripartite information'', which corresponds to
the limit in which the intervals' lengths are infinitely larger than their
(large) distance, supports the claim of universality recently made studying
noninteracting spin chains. Our mild assumptions imply that the residual
tripartite information after a small quench of the anisotropy in the gapless
phase of XXZ is equal to $-\log 2$.
