In the realm of practical application, an increasing number of manufacturers and retailers have embraced the utilization of blockchain technology to augment transparency and traceability within their supply chains. Nonetheless, the optimal allocation of blockchain implementation within a supply chain involving dual channels, while simultaneously addressing consumer privacy concerns, remains a subject of ambiguity. By elucidating this matter, our comprehension of the potential enhancements in supply chain management resulting from the implementation of blockchain technology can be greatly advanced. To this end, we conduct an investigation into the impact of blockchain technology on a supply chain encompassing both retail and direct channels for product distribution. Our approach entails the development of a game-theoretical framework that encompasses a manufacturer and a retailer, each endowed with the autonomy to determine the appropriate utilization of blockchain technology. Specifically, the manufacturer's adoption of blockchain technology allows for its application across both channels (MA), while retailer's adoption is solely to the retail channel (RA). In comparison to a scenario without blockchain (NN), our findings demonstrate that the adoption of blockchain can yield mutual benefits for manufacturers and retailers, irrespective of whether it is implemented under MA or RA. Moreover, we identify the conditions under which each of the three blockchain adoption formats (NN, RA, MA) emerges as an equilibrium solution. Furthermore, we substantiate that each of these blockchain adoption formats has the potential to benefit consumers under specific conditions.