Here it goes.
Mr. Product: I do not know the numbers.
=> Either none of the numbers are prime or one of the numbers is prime but not both. (If both numbers are prime then Mr. Product would just factor the product and have the number.)
Mr. Sum: I knew you didn’t knew [sic] the numbers.
=> The sum can’t be even. (Goldbach’s conjecture – Every even integer greater than 2 can be written as the sum of two primes. Therefore the sum can’t be even or it can be written as the sum of two primes which we know the numbers are not two primes. Sorry everyone but the proof is beyond the scope of this thread.)
=> One number must be odd and one number must be even.
=> The product must be even.
Mr. Product: Now I know the numbers
=> There is one unique way to attain that product using an odd number and an even number. (There is at least one other way, possibly more, to attain that product using two even numbers since one number must be composite.)
Mr. Sum: Now I know the numbers, too.
=> All of us in the CR also do!
4 & 13
There aren’t as many possibilities as you may think once you start throwing out possible pairs based on criteria. The upper bound is “red herring” as it will be the same answer regardless on whether it is 15 or 1,000,000. I tried an upper bound of 10 and found no solution. When I raised it to 15 I found this solution which I believe is the only solution regardless of the upper bound.