It is highly unlikely that the recipient is in the same hospital. The ER team who oversees the brain-dead person is not the same team who handles the organ transplantation. The docs who treat the patient with a failed liver aren't the people who decide where the organ(from the deceased donor) goes. This is a myth that needs to die. Doctors don't 'let someone die' because they are an organ donor. There's a complex system in place to regulate the distribution of organs.

edit: I am a liver transplant recipient.