I don't have insider information, but: if one of the AI companies really wants their models to become really good at this and publicly available datasets are scarce, they can probably just buy anonymized X-ray/MRI scans paired with the human doctor's diagnosis, and train on them. I don't know what the legal story is around this, but AI companies have near infinite money, so I'm sure they can buy their way around regulations (eg. by buying them from a less regulated country).
My understanding is that medical images are part of a patients record (so they must be available for the patient or other docs at the request of the patient) but whoever has collected the images does have some form of ownership. I’m not a lawyer and I have a cursory understanding of this. I believe it would be possible for an AI company to lease or get access to the data through a transaction but I suspect that it hasn’t happened (or happened publicly) due to fear of backlash.
For example I know that some companies like Tempus had access to imaging that corresponded to tumors which had been biopsies for sequencing and they were developing models in house.