This is the case, but waging interstellar war necessarily will reveal your position to the grater galaxy. In which case, you have "poked your head up". If you assume that there are more than two such civilizations then the war continues until there is only one.

At some point there are only two, it is unlikely that both will be eliminated to the point that neither can never rise again.

Run the thought experiment; assume you are civilization C, you detect that somewhere "near by" a civilization (Civ A) sends out a signal. A short time later a great war breaks out. Civ A is utterly eliminated, you detect that there are slagged planets and exploded moons. Whilst you could not detect directly that Civ B; you know that Civ B is out there somewhere, you know the time delay and thus can estimate the probable radius within which Civ B could exist. You step up your passive detection efforts focusing on eventually (100's of years) you find Civ B. Now you know that they are doomed, but you need to ensure you don't meet the same fate. But you also know that another civilization may have done exactly the same thing, and is watching Civ B for any sudden change. But you can't let a know civilization exist when you known that they may find you at any time, and they will eliminate you as soon as they find you.