We present two primal methods to weakly discretize (linear) Dirichlet and (nonlinear) Signorini boundary conditions in elliptic model problems. Both methods support polyhedral meshes with non-matching interfaces and are based on a combination of the Hybrid High-Order (HHO) method and Nitsche's method. Since HHO methods involve both cell unknowns and face unknowns, this leads to different formulations of Nitsche's consistency and penalty terms, either using the trace of the cell un-knowns (cell version) or using directly the face unknowns (face version). The face version uses equal order polynomials for cell and face unknowns, whereas the cell version uses cell unknowns of one order higher than the face unknowns. For Dirichlet conditions, optimal error estimates are established for both versions. For Signorini conditions, optimal error estimates are proven only for the cell version. Numerical experiments confirm the theoretical results, and also reveal optimal convergence for the face version applied to Signorini conditions.