In the UK it's a matter of the patients' family or spokesperson making a statement, sometimes in conjunction with the hospital management. If the media hasn't gotten wind of a given person's admission families often prefer to keep such things quiet, so the usual confidentiality applies.
I ask because while the health of the President of the USA is on its own level, it seemed unusual and a touch theatrical for an entire row of doctors lining up earlier today behind the physician giving the update. Am I right in thinking that kind of presentation is a one-off, or does it happen occasionally in the US and elsewhere? Are there times where management can encourage a patient to allow a doctor to provide an update on their behalf? On the one side there's public interest, on the other there's the media presence being a nuisance for staff and patients.