Much like having different hammy actors in the same work, sometimes putting two snarky characters into the same works and having them interact in some form of discussion (especially an argument) can seem like a competition in sarcasm. This often makes it a little easier to tell which character is more of a snarker; it can also be a sign of Belligerent Sexual Tension, as it's a form of combat in which the antagonists suffer no actual physical damage, and many of the attacks can double as denials of the attacker's true feelings for the target (or be interpreted as such).
See also Volleying Insults and World of Snark. Well, Excuse Me, Princess! is a specific version of this between a snarky hero and a snarky princess. Also may figure in Passive-Aggressive Kombat and Politeness Judo. Bantering Baddie Buddies often engage in conversations like this.
Alternatively could be called "having a sass off".
Not to be confused with cruise missile warfare.