Usage of adverb 'each'

The question I'm asking is why 'they each have their own skills' doesn't become 'they each have his or her own skill,' other than 'they have his or her own skill each.'

Each is a strange word - it functions similarly to, but does not mean the same as, every, all, and both.

The OED does not classify each as an adverb but as a pronoun or an adjective. In this sense, "each = "when considered individually" (MW classes it as all three and I don't think that this is strictly correct.)

To add to this, each is also a determiner or a quantifier, but these act adjectivally:

For example, "Each (determiner or a quantifier) boy had a cold and each (pronoun) had a handkerchief."

Thus 'they each have their own skills' = 'they {when/if [they are] considered individually} have their own skills' and {when/if [they are] considered individually} modifies "they" adjectivally.

It is also clear in "The tickets cost £20 each" = "The tickets cost {£20 when [they are] considered individually.} and {£20 each} becomes a noun phrase, and is no different from "The tickets cost £20."

Thus in "they have his or her own skill each", each is too far removed from its subject, and seems to refer to "skill."