This is one of the biggest debate 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 approach when it comes to training them. Most gym goers who enter the gym for the first time want those arms to swell like that of their Hollywood/Bollywood icons. As a enthusiast without the right knowledge the easiest way to achieve them is to train them more often so that they grow.
The answer to the question depends upon the goals and the purpose with which one trains, also in a lot of cases situations play a vital role in what people think about training arms on separate days. A common trend we observe in the people who do not emphasize on training arms separately are the ones who follow powerlifting/weightlifting training pattern. The reason being powerlifters and weight lifters focus a lot on the structural lifts which are deadlifts, squats, clean jerk, bench presses etc. The primary objective of their training is to improve strength in these lifts and maximise their one rep max, while symmetric physique and aesthetics are not their actual goals . 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. E.g. The triceps are worked very well when you perform your chest and shoulder routine on one day where you do all push exercises together and the biceps alternatively with all the pulling movements. 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 each and 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 the answer can never be proven right or wrong. Each person having 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, but a complete push muscle workout followed by a pull muscle with legs worked separately will ensure a good development of the overall musculature. The smaller muscle groups like the triceps and biceps will still be going through a break down as one trains the larger muscles.