Sorry, not using "Each began..." That would make the whole question moot. Using "They began..." if you please... or even if you don't please. It's the use of "they" that throws it into an odd sphere for discussion.
Yes, I have always used either "each other's" or "one another's". To me, "one" and "each" would make it logical and normal. I can't accept "the view of the other" as anything but the view of some one person in opposition. It's correct to say "He began to understand the other's point of view", or "All of them began to understand each other's point of view" but not "All of them began to understand the other's point of view".