I run it, it works.
I run it with zeros: it works
I run it with double digit numbers - it works.
So what is the problem? The only hassle I can see is that your output are a bit odd: your array collections of odd and even numbers are unnecessary and not good since you only set the indexes where you meet them.
By the way: you don't need three loops: You can use two, or even one.
printf("Enter 10 integers:");
if(value % 2 == 0)
printf("\nSum of odd numbers is %d\n",sum);
printf("\nproduct of even numbers is %d\n",prod);