"Hablé con la persona correspondiente."

We would seldom say "the corresponding person" in English. I don't think that this is a good example to use for "correspondiente" unless this is a common construction in Spanisish.

What does this mean? Is it like a corresponding secretary (that archaic position in organizations where "ladies" dressed in hats and gloves and met for "luncheon"?)

this does not sound right in English but i am at a loss as to what the correct sentence should be. Any takers?

what does that suppose to mean?

Correspondiente also translates as appropriate. 'I talked to the appropriate person' makes more sense in English.