You may cast Double Negative with no targets, one target, or two targets.

In order to counter two target spells, both of those spells must be on the stack at the time you cast Double Negative. Two spells can be on the stack at the same time as the result of an ability (such as cascade or replicate), as the result of a copy effect (such as the one from Cloven Casting or Twincast), or because a spell has been cast in response to another spell.

It’s unusual for a player to cast a spell in response to his or her own spell, and that will be specifically called out when it happens. Under normal circumstances, if a player casts a spell, it is assumed that the player passes priority afterward. If that player casts another spell, it is assumed that each other player has also passed priority, which caused the first spell to resolve and the first player to receive priority again. Unless the first player says otherwise, only one spell will be on the stack at a time in that scenario.