Advances in IT have enabled some firms to offer personalized products according to the private information disclosed by consumers, while others are still offering standardized products, which brings about asymmetric competition. For consumers, disclosing private information for personalized products leads to reduced misfit cost as well as privacy loss. To illuminate the impact of consumers' trade-off between the benefit of information disclosure and the associated privacy concerns on firms' asymmetric price competition, we consider a setting where only one firm is capable of product personalization based on consumers' personal information. The capable firm makes a profit from selling the product and monetizing consumers' information. We demonstrate that as the capable firm becomes more adept at personalization, he may raise or lower the price depending on his profit foci, and an improvement in his capability does not always guarantee a higher profit. Counterintuitively, an increase in the unit misfit cost (i.e., greater product differentiation) can, under certain circumstances, intensity price competition, making both firms worse off and leading to higher consumer surplus. We also show that when consumers are more privacy-concerned, there exists an indirect effect that weakens the impact of an increase in price on the monetization of consumers' information, and hence price competition can be mitigated and both firms can be better off. Furthermore, we demonstrate that product personalization with misfit-reducing effect always increases consumer surplus under the asymmetric competition. Our findings provide firms and policy-makers with great managerial insights.