Same corporations happily use Office 365 and GSuite for all their documents rather than a local and private solution. I'm sure they'll come to a similar arrangement with regard to AI.
That doesn't change the fact that these corporations have an interest in data privacy and cost reduction. To wit, if MS wouldn't guarantee data security in european datacenters, they would be way less competitive in the EU.
Many corporations run their own datacenters, have on premises mailservers and run their own backup solutions. Not everything is in the cloud.
