We study fluctuations of the current at the boundary for the half space asymmetric simple exclusion process (ASEP) and the height function of the half space six vertex model at the boundary at large times. We establish a phase transition depending on the asymptotic density of particles at the boundary, with GSE and GOE limits as well as the Baik--Rains crossover distribution near the critical point. This was previously known for half space last passage percolation, and recently established for the half space log-gamma polymer and KPZ equation in the groundbreaking work of Imamura, Mucciconi, and Sasamoto. The proof uses the underlying algebraic structure of these models in a crucial way to obtain exact formulas. In particular, we show a relationship between the half space six vertex model and a half space Hall--Littlewood measure with two boundary parameters, which is then matched to a free boundary Schur process via a new identity of symmetric functions. Fredholm Pfaffian formulas are established for the half space ASEP and six vertex model, indicating a hidden free fermionic structure.