  We study the full counting statistics for interacting quantum many-body spin
systems weakly coupled to the environment. In the leading order in the
system-bath coupling we derive exact spin current statistics for a large class
of parity symmetric spin-1/2 systems driven by a pair of Markovian baths with
local coupling operators. Interestingly, in this class of systems the leading
order current statistics are universal and do not depend on details of the
Hamiltonian. Furthermore, in the specific case of symmetrically boundary driven
anisotropic Heisenberg ($XXZ$) spin 1/2 chain we derive explicitly the
third-order non-linear corrections to the current statistics.
