Depends on what you want to do. Some task specific models can be trained with a few ten or hundred thousand training examples so you can use a bigger model to produce synthetic training examples and then fine tune a smaller student model. I think that's the usual process. Whether you'd get acceptable performance this way depends, as mentioned, on what you're trying to do and what you'd consider acceptable.

once you are able to get the full probability distributions per token you can distill it on specific domains. distilling without that isn't generally a good idea unless you have invested millions in the requisite infrastructure.