This had me scratching my head for a while. If you don't believe in an all-powerful, all-good God, well, evil is not a "problem" for your position, is it? Atheists don't face the "problem of evil".
But then I realized that, actually, two quite different problems are being run together here.
There is of course a "problem of evil" that atheists face: the problem of how to deal, psychologically, with evil - with the sheer quantity of suffering, and also moral depravity, that so many of us have to endure.
The believer can find solace in their faith, of course. But to what can the atheist turn?
So yes, atheists face this "problem of evil". But it's a different problem. The right response to this move, I think, is to say that the theist has simply changed the subject.
They are trying to suggest, of course, that as the "problem of evil" is just as much a problem for atheists, it doesn't support atheism over theism.
But, as an argument against he truth of theism, the "problem of evil" is not a problem for atheism at all. Vast quantities of evil are evidence against the all-powerful-and-all-good-god hypothesis. They are not evidence against the no-god hypothesis.
True, atheists also face the question - how, if at all, are we to deal psychologically with evil? And it's true that they don't have a pat answer available to them in the way the theist does.
But even if the correct answer is: actually, many of us cannot deal psychologically with such evil without the support of religious faith, that would not give even one ounce of support to the claim that what the faithful believe is actually true.