At the height of the war, could an American diplomat go to Germany to talk? And could a nazi diplomat talk to Washington DC? If not, when did all diplomacy halt?
Follow up question - if there were diplomats located in enemy territory how did they travel to or from their own country without being killed, and how did they communicate with their own government without their communications being intercepted by the enemy?
There seems to have been constant diplomacy through third parties, usually neutral nations. This is true for the UK and the British Commonwealth, not sure about the US. Pretty much any substantial British history of the period will mention that. Subjects discussed included prisoner exchanges and activities with neutrals. Try Winston Churchill's "History of the Second World War".
As for the follow-up question. When Japan launched its surprise attacks on the US, then on Britain and her allies, the diplomats of all countries found them selves in enemy territory but, following international law, were exchanged safely. Michael McKernan's "The Strength of a Nation' ISBN 9781741754452 describes the exchange of Japanese and Australian diplomats which took place in Madagascar.
Most embassies communicated by landline telegraph, (today by the internet) but even today they usually have a shortwave radio transceiver for coded communications. Of course, the enemy was always listening.