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".