The pooling is your signal that something's changed, so start by walking the ditch after the water settles and really look at what's happening. I had a similar setup years back - shared drainage line with overgrown brush, and I kept assuming it needed regrading when really tree roots and accumulated leaves were creating little dams. Before you call someone in to reshape the whole thing, clear out any obvious debris: fallen branches, leaves, sediment buildup. Walk the full length and see if water moves once you've removed that stuff. If it still pools in the same spots after clearing, then you've got a grading problem - the ditch bed has probably settled unevenly, creating low spots where water just sits instead of flowing through.
The maintenance piece matters here too. A shallow ditch needs regular attention, especially if you're getting heavy spring storms. Every year or two, I'll walk mine and pull out what's accumulated. That saves you from needing major work later. For regrading, you'll want to measure the slope if you're serious about it - the ditch should drop maybe an inch or two of elevation for every 10 feet of length, though that depends on your soil type and how fast the water needs to move.
Since this borders your neighbor's property, worth having a conversation before you do anything major. Whoever maintains it (check your property documents) should share the responsibility, and you'll want to be on the same page about whether clearing or regrading makes sense. A lot of disputes come from one person changing the drainage without telling the other.