We study the directed transport of bosons along a one dimensional lattice in a dissipative setting, where the hopping is only facilitated by coupling to a Markovian reservoir. By combining numerical simulations with a field-theoretic analysis, we investigate the current fluctuations for this process and determine its asymptotic behavior. These findings demonstrate that dissipative bosonic transport belongs to the Kardar-Parisi-Zhang universality class and therefore, in spite of the drastic difference in the underlying particle statistics, it features the same coarse-grained behavior as the corresponding asymmetric simple exclusion process for fermions. However, crucial differences between the two processes emerge when focusing on the full counting statistics of current fluctuations. By mapping both models to the physics of fluctuating interfaces, we find that dissipative transport of bosons and fermions can be understood as surface growth and erosion processes, respectively. Within this unified description, both the similarities and discrepancies between the full counting statistics of the transport are reconciled. Beyond purely theoretical interest, these findings are relevant for experiments with cold atoms or long-lived quasiparticles in nanophotonic lattices, where such transport scenarios can be realized.
