Bleg on Counsel in 42 U.S.C. 1983 and Bivens Litigation

I’m presently writing an article for the Cato Supreme Court Review, and I had a question that I hope some readers may help me answer: In terms of counsel, who represents defendants in civil Fourth Amendment cases against government officials in 1983 and Bivens actions, and how is this arranged? Just based on cases I’ve worked on and seen from afar, it seems like DOJ almost always represents the federal officials in Bivens actions, at least barring unusual circumstances; that state lawyers almost always represent state officials in 1983 actions; and that 1983 actions against county or municipal officials are a mix, sometimes involving lawyers for the government and sometimes private firm lawyers hired to represent the officials’ interests. Is that accurate? I’m particularly interested in knowing when the decisions of how to litigate those cases are likely to be made by the government as compared to private counsel hired to represent the officers. (I hope that makes sense — it will make more sense in a few weeks when I can post the draft.) Thanks in advance for any help.