When you choose to engage an enemy, do they engage you too?
Yes.  From page 16 of the rules


  Whether an enemy is engaged through an engagement check, through a
  card effect, or through a player’s choice, the end result is the same,
  with the enemy and the player engaging one another. In all cases, the
  player is considered to have engaged the enemy and the enemy is
  considered to have engaged the player.