Question: Getting clear user feedback is great. But what do you do with the feedback when different people, or groups of people, want different things? 

I have encountered this situation a few times recently.

One example, while testing a new product release: Some users liked the new separate input boxes for x and y coordinates, as the previous version of the software had a single box, with a comma separator. Now they didn't have to type the comma in.

But other users disliked the move to the dual boxes, because it increased the amount of work involved in copying/pasting co-ordinates between boxes. 

I'm not asking for a design solution to this single box vs. dual boxes problem :) I'm asking, more generally, what methodology do you employ to arrive at solutions, where there are two or more opposing feature requests with seemingly equal merit?

Answer: 
Accommodate both! In this case, your users told you exactly what they were missing in the old version (in this example, easy copy/paste). So create a new way that meets both sets of requirements.

Generally speaking, say the old way offered Features A and B, and the new way still supported Feature B, lost feature A, but added feature C. Users said they like C but missed A. So design a widget that offers A, B, and C!

In this example, when the user pastes a comma-separated value, you could automatically transfer the part after the comma to the second box. Also, if they are typing in the first box and type a comma, automatically transfer focus to the second box as they continue typing. You can see this behavior in a lot of phone number entries on forms. Perhaps this would satisfy all your users.
I am programmer learning a bit of UX so I may get beat up by UX purest but I capture metrics on user productivity. If a top performing user asks for a tweak I will take that over a low performing user. I am in an environment where contract data entry use more than one product. Too often a low performing user will state the (my) product is the problem. I will show the customer paying the bill that the top 1/4 users are doing over 1/2 work (and they always are) and those are not the people complaining. I am not saying don't accommodate the low end users. But don't sacrifice the high end users. A user that has performed a task twice is going to have in uninformed opinion. Yes make it more intuitive. But the top performer that says it saves me a key stroke is who wins in my book (as that is what the people footing the bill care about with my product).