Almost every AI project defaults to a public API, because that is the fastest way to see something work. That default is often correct and it is almost never examined. It should be, because by the time a prototype has become a system, three things have quietly become true: every document you process has been transmitted to a third party, your cost scales linearly and permanently with usage, and your service availability is somebody else's operational problem.
None of those is automatically unacceptable. All three are decisions, and they deserve to be made rather than inherited. For a large number of organisations — regulated ones, ones with genuinely confidential corpora, ones running high sustained volume — a different answer is both cheaper and easier to defend.
The honest position is that there are five places your AI can run, they sit on a spectrum from convenient to controlled, and the right one depends on constraints we can work out with you in an afternoon. What follows is that spectrum, without a thumb on the scale — we build all five — and the hybrid of them — and the one that fits you is not the one that is most work for us.