Recent theory results [Walldorf et al., Phys. Rev. B 100 121110 (R) (2019)], obtained in a one loop non-interacting magnon theory, demonstrate a dynamical phase transition in the antiferromagnetic phase of the 2D Hubbard model upon laser driving. The transition is characterized by a qualitative change in the magnon distribution function as the drive strength is varied. Here we investigate the effects of magnon-magnon interactions using an interacting spin-wave theory in a large spin expansion and a Boltzmann formalism. The scattering leads to qualitative changes with respect to the noninteracting results, in particular to steady states that can be characterized by a generalized Bose-Einstein distribution with an effective drive-dependent chemical potential. Implications for the dynamical phase transition and the Mermin-Wagner theorem for nonthermal states are discussed.