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Very differently, though still with some overlap: multiple instinctive approximate models, but then detailed step-by-step models for exact computation. Though the exact model will tend to be somewhat made up on-the-fly.
The first one, I immediately think 8200 plus the sum of 84 & 39. I then work that out (123), and there we are 8323...
That's interesting. I do roughly the same thing: basically add digits left to right, and occasionally correct where I'm at when there's something to carry. But if the numbers were on paper I'd probably go right to left. 
I'd do very different things with other 4 digit numbers. In this case it immediately seems to me that 58+24 and 84+39 are likely to be easy, so this is a good route. With other numbers I'd use different routes.
Which is a long way from the grade-school algorithm, I guess. There's certainly some sense of making the algorithm up on-the-fly, in response to a sense of what will work most easily. (And sometimes messing up and backtracking when I realize a different approach would be easier)
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