A co-respondent is a person named in an old-style divorce case because they are said to have had sex with the husband or wife of the person asking for the divorce. In that kind of court story there is the one who wants the divorce, the husband or wife on the other side (the respondent), and the third person (the co-respondent). It is a law word, and it feels dated: many places have moved to divorce that does not need this kind of proof, so you no longer have to name a lover in the papers. You will still meet it in older novels, newspapers, and legal history. It is a cold paper word for a painful private thing. Though actually it is not a word for a news writer. That is correspondent — extra letters in the middle, different job.