So, to clarify and build oon what the above two teachers have said: The sentence as it stands is incorrect. You would have to add 'from the college down the road' in order to make the sentence correct.

Here you are students of English in general. We don't use 'the' when we are talking about things in general. You could make it concrete 'the' by being more specific. "We are the English students from the college down the road."

yes you need to include 'the students'. 'the' specifies the students that are not there the specific students you are enquiring about. "THE students that should be in their class" either your class THE class you are going to give or start now/soon. "THE students that should be in the class/classroom" THE class they enrolled for or are expected to be located at. P.S AND to remove any ambiguity. The sentences is incomplete and what you are trying to say is unclear. "We are students of English" is correct if stating what you are students of. "we are English students" = Students studying English or English people citizens studying X [we don't know] The sentence you provided is unclear as to whether you mean you are students or you are THE students OF teachers Karen and Bob.