SandandSea there are different answers depending on which donation you are asking about. Legally, and technically, though, the answer is yes, no matter which donation you are talking about. I'll take you through each, to explain further, as it is a bit complicated.
Donation after Circulatory Death (DCD): A patient will donate organs as soon as they have been declared dead (defined by the absence of a central pulse on palpation or heart sounds on auscultation for a period of 5 minutes), by being transferred to theatres. They will be legally dead from the time declared by the doctor who observed them to have no circulation.
Donation after Brain Death: A lengthy and complex set of tests is carried out at the beside, which tests all cranial nerves for signs of reflex responses. There are 12 cranial nerve pairs, and each test intends to stimulate these nerves. The test is carried out by two senior doctors, often observed by others, and if any person sees a response they say so. If no response is seen, the patient is declared brain dead at the conclusion of the tests. The tests are repeated a second time for confirmation, but the time of death is the conclusion of thefirst set of tests. If the patient is going to go on to be a donor, they will remain connected to a ventilator, and treatment will continue, despite being legally dead, so that their organs can be in the best state possible for transplant. They will go through to theatres connected to the ventilator. Legally, technically and for all practical purposes, however, they are dead. They have been proven to have no brain function. Everything that made them who they were has, sadly, gone. They will never wake up. They will never think, decide, choose, question, respond, be happy, sad, angry, cold, hungry, again.
The last donation type is tissue donation. Patients can donate bones, tendons, skin, corneas, etc., and to do that they just have to arrive at the mortuary within a few hours of death. So they are are definitely dead then when they donate, also.