There's a misconception in the question that is important to address first: when an LLM is running inference it isn't querying its training data at all, it's just using a function that we created previously (the "model") to predict the next word in a block of text. That's it. When considering plain inference (no web search or document lookup), the decisions that determine a model's speed and capabilities come before…
> But this is because larger models tend to be better, not because throwing more compute at an existing model helps it produce better results. There's a caveat here - allowing the model to produce more tokens (i.e. giving it more compute time to "think") can produce better results. E.g. asking a model to reason before producing an answer, leads to better answers. And the extra tokens = more compute.
The concern people might feel when they realise an ai might have private thoughts is another issue entirely.