Is the second free machine in a "buy one, get one free" offer considered permissible (halal) or forbidden (haram), and is the money earned from using it for work permissible (halal) or forbidden (haram)?

This sale is permissible. It is a reduction in the price of the commodity, selling it for half price, and it is permissible. There is no disagreement among contemporary jurists that a clear gift attached to the commodity is permissible, because it is like a reduction in its price. Therefore, there is no harm in buying two machines for the price of one, and you may benefit from each of them for any permissible purpose, and the money earned from them is lawful.