In the case of the fifth century BC (the era of the Peloponnesian Wars) a lot of diplomacy would be conducted through proxenoi who were citizens of city state A but were bound by ties of guest friendship to city state B. A proxenus would be responsible for supporting friendship with city B in his own city and would also have duties to protect citizens of city B traveling in city A, sort of like a modern consul. There were also delegations appointed on an ad hoc basis as with the Spartan delegation to Athens dispatched after the Spartans were defeated at Sphacteria near the end of the Archidamian War (the first part of the Peloponnesian War describes by Thucydides). At the lowest level negotiations between individual military units in the war would be conducted with the aid of heralds under the protection of hermes, but this would normally be limited to truces over the recovery of bodies (recovering the bodies of your dead soldiers under truce was seen as an admission of defeat) and occasionally negotiated surrenders.

Source: Thucydides’ History of the Peloponnesian War

Edit: In more specific answer to your question, all the major city states would have had a proxenus in the other major city states at all times, the appointment of delegations at least from Thucydides’ History seems rarer and only to be employed in response to some sort of crisis (after Sphacteria the Spartan delegation was attempting to secure the release of several hundred upper class prisoners of war which was a major political issue in Sparta) or as part of peace negotiations as with the peace of Nicias that ended the Archidamian war. Lastly the herald system seems to have been employed regularly after battles for the sorts of truces I described, but there doesn’t seem to have been too much contact between opposing units in the field. If you’re interested in city state relations in Ancient Greece I would also definitely recommend reading a little bit about Xenia or the Greek guest-host relationship which runs through all of the fabric of Ancient Greek culture and is helpful in understanding everything from the odyssey and Aeschylus’ suppliants to the sorts of diplomatic agreements you were asking about (and as you might have guessed is the root of the word proxenus).