We study a mixed duopoly model, in which a state-owned (public) firm maximizing domestic social surplus and a private (foreign) firm compete. Under general enough assumptions, we first justify the concept of conjectural variations equilibrium (CVE) applied to the model by demonstrating concavity of the expected profit function of each agent. Next we establish existence and uniqueness results for the conjectural variations equilibrium in the described duopoly. A particular case of a CVE with constant influence (conjecture) coefficients is also examined.