Ahh, I love double triggers. Instant choice between two chokes, or between two loads. The whole point is you can choose based on the shot that is presented.
With that out of the way, I believe you will find most SXS's are set up with the more open choke connected to the front trigger, and the more closed choke connected to the rear trigger. (If you have fixed chokes, anyway.) That's how mine is set up. So I suppose that means that you ordinarily shoot the front trigger first, since the choke is more open and the bird is--theoretically, at least--closer to you. The second shot will likely be farther away, so you use the tighter choke, which is the rear trigger.
When I hunt upland in the fall, I run into a mixed bag of doves and Hungarian partridge, which are slightly larger. I tend to load 6's in the open choke tube, and 5's in the closed choke tube. If I flush a dove, its the 6's on the front trigger. If I flush Huns, its its the 5's on the rear trigger. Try that with your pump gun!
Longest shot I ever made on a dove was about 45 yards. He was moving straight away--with the bead right on him he was not moving at all. So I hit him with the 5's on the rear trigger, and down he went.
I'm a big fan of twin triggers, if you can't tell...