Would a population benefit from a requirement that all elected representatives publish their administrative correspondence, i.e. all emails from one civil servant to another? What would be the costs or benefits of this kind of policy?
Especially when it comes to elected positions, they are political actors who negotiate outcomes with other political actors. In this, they will have allies and enemies. If allies cannot coordinate their positions, the negotiation process breaks down.
There has to be a process to scrutinize communications after the fact, when there is suspicion of inappropriate actions, but a state governor should be able to write to a legislator "ideally we want $80M for this program, but we're prepared to give it up if we can get the improved pollution standards instead."
One obvious benefit is transparency. You would know at all times what your representative is saying in official correspondence, which could later be used to catch them in wrongdoing. Many political scandals in the U.S. came to light as a result of the Freedom of Information Act (FOIA), under which any citizen can demand the official communications of most government officials. To pick one example off the top of my head, the New Jersey "Bridgegate" scandal had legs because a FOIA request caught the governor's staffer texting another official to create unnecessary traffic problem.
But the downside-- and the stated reason the U.S. Congress exempts itself from FOIA-- is to quote the old axiom, "Laws are like sausages. It's better not to see them being made." Legislative activity means politics, and politics means deal-making, and such deal-making is often awful in the light of day. Important discussions and trade-offs might not occur if legislators know that they cannot speak frankly with each other without it later being used against them.
Requiring disclosures of official communication could also backfire by leading politicians to resort to unofficial communication, which in turn would harm historical record-keeping and transparency. Hillary Clinton was subject to FOIA requests as a cabinet member for example, and while she insists she didn't set up her private sever intending to avoid turning over documents to journalists and Congress... well, that's exactly what ended up happening.