"Assignee": doctor or patients

Solution 1:

If we think of assignee as someone who has been assigned a job, it is the doctor who should be the assignee and not the patient, as it is the doctor's job to see the patient and not the patient's job to see the doctor.

Solution 2:

The assignee is the person to whom the assignment is made. Since you said "Each doctor will be assigned to check on..", it sounds like the patients are assigned to the doctor. In that case, the doc is the assignee and the patient is the assignment.

You could express the relationship in the other direction, by assigning doctors to patients, in which case the patient is the assignee.

Solution 3:

The assignee is the person who has been assigned.

http://www.oed.com/view/Entry/11916?redirectedFrom=assignee#eid

The assigner makes the assignment, and the assignee is the victim. In the OPs situation, the doctor is the assignee.

But...

...the OP is asking whether the assignees for the patients is the doctor, or the assignees for the doctor are the patients. This may be a false question, as the the patient (I assume) has received no assignment. The doctor is the assignee for whoever did the assigning.

The confusion is that the doctor is assigned to a patient, but that doesn't alter anything: the patient didn't do the assigning, nor did the doctor.

Solution 4:

The problem is in your second sentence. Assign can mean allot; designate; transfer; specify; or archaically mark out. But in every case it is transitive; you assign a noun to somebody. Myself, I would say each doctor was assigned a list of patients (= a list of patients was assigned to the doctor), in which case the doctors were the assignees.It would be equally possible to say each doctor was assigned to a list of patients, in which case the patients are the assignees. But you have to be clear about using the verb before this question can be answered properly.