Cloud-hosted Large Language Models (LLMs) offer unmatched reasoning capabilities and dynamic knowledge, yet submitting raw queries to these external services risks exposing sensitive user intent. Conversely, relying exclusively on trusted local models preserves privacy but often compromises answer quality due to limited parameter scale and knowledge. To resolve this dilemma, we propose Game-theoretic Trustworthy Knowledge Acquisition (GTKA), a framework that formulates the trade-off between knowledge utility and privacy as a strategic game. GTKA consists of three components: (i) a privacy-aware sub-query generator that decomposes sensitive intent into generalized, low-risk fragments; (ii) an adversarial reconstruction attacker that attempts to infer the original query from these fragments, providing adaptive leakage signals; and (iii) a trusted local integrator that synthesizes external responses within a secure boundary. By training the generator and attacker in an alternating adversarial manner, GTKA optimizes the sub-query generation policy to maximize knowledge acquisition accuracy while minimizing the reconstructability of the original sensitive intent. To validate our approach, we construct two sensitive-domain benchmarks in the biomedical and legal fields. Extensive experiments demonstrate that GTKA significantly reduces intent leakage compared to state-of-the-art baselines while maintaining high-fidelity answer quality.