Should you structure your argument in an essay by writing one side and then the other, or write it paragraph by paragraph?

Both of these are acceptable, but I would advise writing it paragraph by paragraph, as quite often students run out of time in the exam and that way at least you've covered both sides of the argument. It also demonstrates your ability to develop a well thought out and complex argument, as it's easier to compare and contrast your arguments and link them more closely, and this will show the examiner that are capable of making interesting links.