The $K$-theoretic Schur $P$- and $Q$-functions $GP_λ$ and $GQ_λ$ may be concretely defined as weight generating functions for semistandard shifted set-valued tableaux. These symmetric functions are the shifted analogues of stable Grothendieck polynomials, and were introduced by Ikeda and Naruse for applications in geometry. Nakagawa and Naruse specified families of dual $K$-theoretic Schur $P$- and $Q$-functions $gp_λ$ and $gq_λ$ via a Cauchy identity involving $GP_λ$ and $GQ_λ$. They conjectured that the dual power series are weight generating functions for certain shifted plane partitions. We prove this conjecture. We also derive a related generating function formula for the images of $gp_λ$ and $gq_λ$ under the $ω$ involution of the ring of symmetric functions. This confirms a conjecture of Chiu and the second author. Using these results, we verify a conjecture of Ikeda and Naruse that the $GQ$-functions are a basis for a ring.