In both the examples above, the argument and its rebuttal are separate. They are made by different people at different times.
But you can, when making an argument, anticipate other people's rebuttals and include them in your argument.
You might do this to limit your argument, explore the evidence, or respond to objections in advance (and make a counter-counter argument).
For example, the New York Times article we snipped above presents claims and rebuttals from both sides of the argument: