This paper studies demand information sharing in a two-echelon supply chain consisting of one manufacturer and two retailers, an informed retailer and an uninformed retailer. The retailers engage in Cournot competition or Bertrand competition. The manufacturer can conduct cost reduction, while the informed retailer has the decision power to opt for a sharing format, among no sharing, partial sharing (only with the manufacturer), and full sharing. Under partial sharing, the uninformed retailer infers the demand information from the manufacturer's wholesale price. We find that the inference effect, that is, the presence of the manufacturer's pricing distortion, depends on the manufacturer's cost-reduction efficiency and the retailers' competition mode. Under low cost-reduction efficiency, the inference effect makes the retailers worse off in quantity competition, whereas it is beneficial to them in price competition. On the contrary, under high efficiency, the retailers benefit from the inference effect in quantity competition, whereas they become worse off in price competition. By contrast, the inference effect only makes the manufacturer worse off. The uninformed retailer may prefer partial sharing to full sharing, while the informed retailer may be indifferent between full sharing and partial sharing.