I've come across the sentence below:
Indiana is one of a few states where students who attend schools separated by county lines, no more than a few miles apart, spend seven months a year living in different time zones.
I think the sentence "students who attend schools separated by county lines" needs the word were (before the word "separated").
I think "students" is the object, but maybe "schools" is the object. I am not sure.
Could you please explain it to me?
Indiana is one of a few states where students who attend schools [separated by county lines, no more than a few miles apart], spend seven months a year living in different time zones.
The bracketed constituent is a past-participial clause modifying "schools".
Past-participials (and gerund-participials) as modifiers of nouns are semantically similar to relative clauses: compare schools which are separated by county lines.
They are not, though, analysed as relative clauses since there is no possibility of them containing a relative phrase (cf. * schools which separated by county lines.)
The objects of the relative clause verbs "attend" and "spend" are respectively schools separated by county lines, no more than a few miles apart, and seven months a year living in different time zones.
Other examples of adjectival phrases used to describe nouns:
Young children driven by car are not in danger of being run over.
The driver of the bus repaired today says it still does not run well.
Hundreds of people dressed in costumes paraded down the street.
They delivered packages prepared in the factory to the train station.
Every one of those imply "which or who were".
The phrases in bold modify the nouns preceding them. They are adjectival phrases.
Let's look at just the clause in question. It happens to be a relative clause:
. . . who attend schools separated by county lines . . .
This clause can be separated into subject and predicate. The subject is the relative "who", referencing students. The predicate is the rest.
Within that predicate we find the finite verb "attend" and its object "schools separated by county lines". The word "schools" is a common noun and the phrase "separated by county lines" modifies it.
The form "separated" could be the past-tense form of the verb to separate, or it could be a participial form. In this case, it is the so-called past-participle form. That is a non-finite form which does not create a predicate and does not demand a subject.
Adding the word "were" where you suggest breaks the clause.
The form "were" isn't a participle. It's finite. It forms a predicate that demands a subject. The phrasing "schools were separated by county lines" can stand as a sentence on its own, with "schools" acting as that subject.
That would leave nothing that can then act as the object of "attend".
If you feel you must add a "were" (or, more appropriately given the surrounding tense, an "are") then you must also add a subject for that finite verb:
. . . who attend schools that are separated by county lines . . .
In this case, we have a sensible subordinate clause which modifies the simple object of "attend". In turn, this matrix subordinate clause can still sensibly modify "students".
