If there is a single noun phrase referring to several things, it's plural. If there are several noun phrases, each referring to one thing, each of the noun phrases is singular. So, it's all about noun phrases, and how many of them there are.
An article starts a noun phrase, so in straightforward cases, you can count the articles to tell how many noun phrases there are. (Some noun phrases don't have articles.) In "Everyone has a left and a right arm" there are two articles in the object, so there are two noun phrases coordinated: [[a left (arm)] and [a right arm]]. In "The left and right arms are both necessary" there is just one article in the subject noun phrase, so there is just a single plural noun phrase, whose noun head is modified by a coordination of two adjectives: [the [left and right] arms], and this is why that noun phrase is plural and its head noun "arms" must be plural.
In ordinary cases of coordination, two things of the same category are connected by "and" to form a new phrase of that same category. That's why when two noun phrases, each with its own article, are connected with "and", the result is a noun phrase, and why when two adjectives are connected with "and", the result is an adjective. People with training in traditional grammar may resist calling a multi-word constituent like "left and right" an adjective, but they ought to get used to it.