It is the prophets who are saying "Peace, peace". It is one of their faults. As a complaint against them it works best if the point is that they are falsely promising "peace" to the people.
The most immediate threat to peace is the literal warfare being predicted in the first twelve verses of the chapter. But there is also a connection between "peace" and "living in obedience to God", as we learn from Isaiah; "O that you had hearkened to my commandments! Then your peace would have been like a river and your righteousness like the waves of the sea... There is no peace, says the Lord, for the wicked" (Isaiah ch48 v18, v22 RSV).
I suggest that the false prophets are offering a double message of comfort; You are in peace with God (he is not angry with you) and therefore there will be no war.
Regarding your second sentence, the AV shows better than modern translations the early arrival of the standard greeting "Is there peace with you my brother?" or something similar. The earliest example seems to be Moses and Jethro asking each other of their peace in Exodus ch18 v7