Can employers from the same corporation, but different branches, discuss a potential employee?

Yes, within certain limits.

Of course, why wouldn't they? If an employee is seeking a transfer or applying for a new job after having previously worked for a sister company, it's SOP to contact the former supervisor.

As long as there are no lies about the employee, they are free to discuss the applicant with respect to work issues.

Yes, and you say potential employee? so does this person work for the other branch (not that it really makes a difference)

But a company is free to search and ask and find information out from any sourse they wish.