I think Joseph's answer is great and works with all such systems. But I happen to be familiar with the ways the Egyptians and Sumerians did their early calculations, and I think it is relevant in the sense that it is also a non-positional number system.
Egypt's number system is both non-positional and base 10 (that is, centered around powers of 10) like the Roman system, but it doesn't have the interesting feature where 4 is IV instead of IIII (using Roman Numerals to represent the Egyptian numbers as well). Egyptian scribes would divide by using a sort of mixture of guess and check, and repeated squaring.
It's easy to see through an example. Suppose we were to divide 153 by 9. Then we write down 1 and the divisor, and double both sides until the right side is bigger than 153.
153 / 9 \\
1 \quad &;\quad 9 \\
2 \quad &;\quad 18 \\
4 \quad &;\quad 36 \\
8 \quad &;\quad 72 \\
16 \quad &;\quad 144 \\
32 \quad &;\quad 288
Now we stop as $288 > 153$. The next step is to represent 153 by adding values on the right (there is only one way to do this). This is where I imagine they used guess and check - they would take the largest number (144 here) and keep adding the next-largest one that doesn't make it too large. 144 + 72 is too big, so we don't use 72. 36 and 18 are also too big. But 9 is just right.
So since we know that $144 + 9 = 153$, we can take their respective 2 powers to see that 153 divided by 9 is $1 + 16 = 17$. And so it is. That's pretty interesting, I think.
I also imagine that if one were calculating things all the time, they'd get really good at using 2 powers and so wouldn't even have to think about it. I also happen to know that they would sometimes guess a number early on and then just work with the difference. This is a lot like Joseph's answer: the Egyptians might guess 20, for example. Then they'd see that $20 \cdot 9 = 180$ and $180 - 153 = 27$. So then they'd just find 27 divided by 9 and subtract it from 20. But again, they would use 2 powers.