Second person or third person?

third person plural: They (Officers) were.

When you use "I" or "we" as the subject, you have the first person. When you have "you" as the subject, you have the second person. When you use "he," "she," "it," or "they," you have the third person.
"Officers" is equivalent to "they"; therefore it is third person. Even if you are referring to yourself as one of the officers, the subject is still third person, grammatically.

Oh wow, Hitch, we responded at the same time. Weird...

You are correct! However, back sixth grade, I had trouble understanding 'we' being in first person. If you think about it, we means more than one. IE We are going to the movies. It could refer to you and your friend, you and your brother, you and your girlfriend, you and your probation officer, you and your grandmother, you and your grandfather, you and your mother and you and your father and etc

we, is first person PLURAL ... as opposed to i, which is first person SINGULAR

as for the original sentence, to include oneself you might say

we, the officers, were dispatched ...

or maybe

officers, myself included, were dispatched ...