He sat in the back of the car with a policeman on either side.
-Longman Dictionary-
Unfortunately I was sitting at the table with smokers on either side of me.
-Cambridge Dictionary-
They both describe two people sitting on either side. But why does Longman use singular and cambridge plural?
Professor McGonagall opened the classroom door at that moment, which was perhaps lucky; Hermione and Ron were looking daggers at each other, and when they got into class, they seated themselves either side of Harry, and didn’t talk to each other all lesson.
-Harry Potter and the Prisoner of Azkaban-
They seated themselves (Ron and Hermione) either side of Harry. Why is it plural (themselves) here? Does it mean they sit on the left of Harry at the same time they sit on the right of Harry.