This is one of the biggest debates that a lot of coaches and trainers have battled for in the past decade. Also, arms being the most favored body part purely for its aesthetic appeal does have a lot of biased approaches when it comes to training them.
The answer to the question depends upon the goals and the purpose with which one trains. People who do not emphasize training arms separately are the ones who follow the powerlifting/
The primary objective of their training is to improve strength in these lifts and maximize their one-rep max. As arms are the assisting muscles in every push and pull exercise, their growth is a by-product of a good explosive workout that happens on the bigger muscles of the body like in the case of weightlifters and powerlifters. So the need for a separate arm training does not arise for such lifters as it becomes a wasted training period for them.
On the other hand, bodybuilders train with the motive of muscular hypertrophy with a motive to create the perfect symmetrical physique that is stage-ready.
Therefore, a lot of bodybuilders along with their bigger body parts also train arms separately just to add that extra pound of muscle from those isolation movements as bodybuilding requires every body part to be developed in symmetry and to your potential max.
This also gets us to the question if even the bodybuilders need a separate day to train arms?
Well, each person has a different capacity to train and recovery will chalk a plan as per his capabilities.
The training pattern may include bigger muscle groups trained together followed by a 2-day break where the smaller muscle groups will be trained. Some bodybuilders follow the push and pull routine where they train all pushing muscles (Chest, Shoulder/triceps) on one day and the pulling muscles(back/biceps) on another day followed by the leg workout. So, the ideal routine is yet to be defined or maybe it never will be.