Compared to traditional machine learning models, recent large language models (LLMs) can exhibit multi-task-solving capabilities through multiple dialogues and multi-modal data sources. These unique characteristics of LLMs, beyond their large size, make their deployment more challenging during the inference stage. Specifically, (i) deploying LLMs on local devices faces computational, memory, and energy resource issues, while (ii) deploying them in the cloud cannot guarantee real-time service and incurs communication/usage costs. In this paper, we design a local-cloud LLM inference offloading (LCIO) system, featuring (i) a large-scale cloud LLM that can handle multi-modal data sources and (ii) a lightweight local LLM that can process simple tasks at high speed. LCIO employs resource-constrained reinforcement learning (RCRL) to determine where to make the inference (i.e., local vs. cloud) and which multi-modal data sources to use for each dialogue/task, aiming to maximize the long-term reward (which incorporates response quality, latency, and usage cost) while adhering to resource constraints. We also propose M4A1, a new dataset that accounts for multi-modal, multi-task, multi-dialogue, and multi-LLM characteristics, to investigate the capabilities of LLMs in various practical scenarios. We demonstrate the effectiveness of LCIO compared to baselines, showing significant savings in latency and cost while achieving satisfactory response quality.