Is Shanghai Pharmaceutical Group a state-owned enterprise or a private enterprise?

Neither. According to the website of Querying Credit China, the enterprise type of this company is other joint stock limited companies (listed). "Other joint stock limited company (listed)" means that the company is not a state-owned enterprise, and the state-owned enterprise is called "joint stock limited company (listed, state-controlled)". The term "other" means that it does not belong to any kind of foreign capital, state-owned or private holding, that is, the equity is relatively dispersed. A joint-stock company refers to a company with shares as its capital, and its shareholders are liable to the company to the extent of the shares subscribed by them.