Cloud APIs bill per token, with separate rates for input (what you send) and output (what the model writes), and much higher rates for the most capable models. Most providers also offer lower rates for batch jobs that can wait and for repeated prompt prefixes that can be cached. See the OpenAI, Anthropic and Gemini pricing pages. Your bill is roughly requests times tokens per request times rate, so long documents and chatty agents cost more than short classifications.
On-premise costs are mostly fixed: the GPU server or workstation, electricity and cooling, rack or office space, and engineering time to install and maintain the serving stack (llama.cpp, Ollama and vLLM are the common choices, depending on the load). The main sizing question is GPU memory. As a rough guide, an 8 GB GPU like the RTX 3070 Ti laptop card in our lab runs 7 to 8 billion parameter models at 4-bit quantization; 24 GB cards handle models in the 30 billion range; 70 billion parameter models need around 40 GB or more at 4-bit, usually across two GPUs. Our GPU servers and AI hardware service sizes this to your workload.
The break-even is not a universal number. It depends on how many tokens you process per month, which cloud model you would otherwise use, and whether a smaller local model is good enough for the task. We work it out from your real volumes during a consultation.