While Google recommends that webmasters utilizes responsive web design, as it also perfectly fine for users to implement a m.example.com mobile URL strategy.  But this can sometimes result in links being split between two URLs which are essentially the same thing, just one is for desktop while the other is for mobile.

How does Google handle this for ranking purposes when links are split?  Does each page get calculated separately and ranked accordingly or does Google combine the two as if they are the same?

The answer is yes, Google will combine the two separate URLs for calculating for PageRank & ranking purposes.

The way to think about it is that if there are two URLs representing the same piece of content (say, desktop and mobile), we’ll try to think of those two URLs as a single document in ranking, so links to either URL will count as a link to the combined document. So if the mobile version has 2 links and the desktop version has 3 links, we have a document with 5 links. When a user searches, we’ll rank that document as having 5 links, and then choose the correct URL to represent that document in search results based on whatever device the user is using.

So as long as you are correctly matching the two URLs together with annotations either on-page on in sitemaps, your links will be attributed from both pages as if it was a single page.