After eliminating all such numbers A knows what number it is, because he knows the sum of the digits. In other words, there is one sum with only one number left corresponding to that sum. That number is 20. B sees that as well and knows the number too.
Unless I have misunderstood the last part of this paragraph: "there is one sum wiht only one number left corresponding to that sum", I believe that it is inaccurate. The idea behind that sentence seems correct, but in reality, both the sums 2 and 17 have only. Two can be made of 20 and 11, 17 can be made of 98 and 89, and unless they hinted to each other what the sum and product were, I cannot see how they would know for certain that 20 was the answer.
I suspect that I have misunderstood the riddle itself in some way, because if I hadn't, I cannot see how you guys could have overlooked what I have just pointed out. If I did misunderstand the riddle, please let me know.