Should the sentence start with "Neither residents nor their guests are permitted to smoke…." ?

Yes.
or

"Residents and their guests are not permitted to smoke...."

It's common now a days to use a sentence like this. It should be "neither residents nor their guests are permitted to smoke." But 'neither' is dropped here and is quite common now a days.

" 'neither' is dropped here and is quite common now a days."

In what sub-culture is this common, acceptable English?

Also:
nowadays