>!10 times.
The following table gives the numbers of one person and the corresponding possible numbers of the other person.
own other
0 5, 8, 15
1 4, 7, 14
2 3, 6, 13
3 2, 5, 12
4 1, 4, 11
5 0, 3, 10
6 2, 9
7 1, 8
8 0, 7
9 6
10 5
11 4
12 3
13 2
14 1
15 0
Round 1:
> If one of them has a number greater than 8 he/she would know that the sum is 15 as it can't be 5 or 8 anymore.
> For the other numbers the following table is true(after both said no):
>> own other
>> 0 5, 8 -> no 15 as the other person would have known it
>> 1 4, 7 -> no 14 as the other person would have known it
>> 2 3, 6 -> no 13 as the other person would have known it
>> 3 2, 5 -> no 12 as the other person would have known it
>> 4 1, 4 -> no 11 as the other person would have known it
>> 5 0, 3 -> no 10 as the other person would have known it
>> 6 2 -> no 9 as the other person would have known it
>> 7 1, 8
>> 8 0, 7
Round 2:
> If one of them has a 6 he/she now knows the number of the other one
> For the other numbers the following table is true(after both said no):
>> own other
>> 0 5, 8
>> 1 4, 7
>> 2 3 -> no 6 as the other person would have known it
>> 3 2, 5
>> 4 1, 4
>> 5 0, 3
>> 7 1, 8
>> 8 0, 7
Round 3:
> If one of them has a 2 he/she now knows the number of the other one
> For the other numbers the following table is true(after both said no):
>> own other
>> 0 5, 8
>> 1 4, 7
>> 3 5 -> no 2 as the other person would have known it
>> 4 1, 4
>> 5 0, 3
>> 7 1, 8
>> 8 0, 7
Round 4:
> If one of them has a 3 he/she now knows the number of the other one
> For the other numbers the following table is true(after both said no):
>> own other
>> 0 5, 8
>> 1 4, 7
>> 4 1, 4
>> 5 0 -> no 3 as the other person would have known it
>> 7 1, 8
>> 8 0, 7
Round 5:
> If one of them has a 5 he/she now knows the number of the other one
> For the other numbers the following table is true(after both said no):
>> own other
>> 0 8 -> no 5 as the other person would have known it
>> 1 4, 7
>> 4 1, 4
>> 7 1, 8
>> 8 0, 7
Round 6:
> If one of them has a 0 he/she now knows the number of the other one
> For the other numbers the following table is true(after both said no):
>> own other
>> 1 4, 7
>> 4 1, 4
>> 7 1, 8
>> 8 7 -> no 0 as the other person would have known it
Round 7:
> If one of them has a 8 he/she now knows the number of the other one
> For the other numbers the following table is true(after both said no):
>> own other
>> 1 4, 7
>> 4 1, 4
>> 7 1 -> no 8 as the other person would have known it
Round 8:
> If one of them has a 7 he/she now knows the number of the other one
> For the other numbers the following table is true(after both said no):
>> own other
>> 1 4 -> no 7 as the other person would have known it
>> 4 1, 4
Round 9:
> If one of them has a 1 he/she now knows the number of the other one
> For the other numbers the following table is true(after both said no):
>> own other
>> 4 4 -> no 1 as the other person would have known it
Round 10:
> If they make it up to this round they now definitely know what number the other person has and at the same time what the sum of their numbers is.
____
EDIT:
This is actually a kind of derivation of the blue eyes problem:
https://puzzling.stackexchange.com/questions/236/in-the-100-blue-eyes-problem-why-is-the-oracle-necessary