This paper explores the equilibrium market outcomes in the contexts of both quantity-setting and price-setting private duopolies with the consistent conjectures of two private firms, wherein they maximize the weighted sum of their own profits and their respective opponent firm’s profit. Similar to the private duopoly without network effects wherein the two private firms maximize their genuine relative profits, in the private duopoly with network effects such that both firms maximize the weighted sum of their own profits and their respective opponent firm’s profit, we show that the equilibrium outcomes in the quantity-setting competition with the consistent conjectures of both firms are equivalent to those in the price-setting competition with the consistent conjectures of both firms.