There are indeed such exchanges. Usually the leader calls. I will, however, give you the opposite example.
In 2010, in the United Kingdom, the outgoing government did not have a plurality of seats. The Conservatives had the plurality of seats but not a majority. There were negotiations with the Liberal Democrats over who would form the government.
Given the adoption of the manual and the discussions that took place, the media held back. None of the British media that evening announced who would form the next government, although there were good reasons to believe that it would be the Conservatives, since they had obtained the most seats, although not the majority. The media gave the parties the time they needed to discuss things among themselves. There were negotiations between the Labour Party and the Liberal Democrats on the one hand, and between the Liberal Democrats and the Conservative Party on the other.
Keep in mind that during the election the Conservatives and the Liberal Democrats had said they would not form a coalition. However, given the results that night, all the parties concluded that it would be in everyone's best interests to form an official coalition rather than having a minority Conservative government.
This was possible in part because, to some extent, the media let the political parties handle the situation and talk amongst themselves. If that night the media had rushed to announce that Party x was going to form government if the trend held, any other party that tried to form government would automatically be accused of fraud or of attempting a political coup.
That is why it would be useful to have such mechanisms in a context where there are more minority governments, to allow elected politicians to decide among them who will form the next government.