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: 
Try to distinguish between *what* users want and *how* they want it done.

Taking your example above, users wanting one vs. two input boxes is all about the *how*. The *what* is being able to paste comma-separated coordinate pairs vs. not having to press comma. (Or, for some users, being able to simply press comma rather than having to click a second input box.)

In the above example you could resolve that in the following way:

* Implement two separate boxes.
* When a comma is entered in the first box, transfer focus to the second box
* Implement copy and paste buttons that will copy the two values to the Clipboard (as comma-separated values) and paste a comma-separated pair from the clipboard, splitting it correctly between both boxes.

Implementations will, of course, vary heavily depending on the situation you're facing. However, many apparent conflicts between requirements will go away when you look at the what/why rather than the how.
User Experience isn't about taking exclusive decisions, is about make the overall experience for the majority of users, the best as possible. I will use your use case as an example. 

**Previous version**
Single text field, coordinates separated by comma.

**Advantages:** copy/paste is smooth

**Disadvantages:** readability

**New version**
Two text fields, one per coordinate.

**Advantages:** Easy to read, UI cleaner

**Disadvantages:** copy/paste is trivial and duplicate steps 

**What is the solution?**

The union of the advantages and the exclusion of the disadvantages. You keep two fields, on each field you intercept the JavaScript event "copy" and "paste" and whenever it happens, you "tweak it".

For example, let's call the fields, "field X" and "field Y", if someone select field X and copy it, with JavaScript you create a string that join the value of 
"field X" and "field Y" and put that in the clipboard, so when user paste it, it would be pasted the coordinates X,Y rather than the single one.

For paste the same, if he paste on X, you split the string if contains an comma and accommodate. So in this case everybody wins.

I could think in more use cases and this could grow, but my point is that you should not focus in solutions as one way, you should focus in have something that satisfied most of the common cases.