Its been a long time since I did any fluids, and even then, it was not much. But I'll take a shot just from simple logic.
Suppose we have a single fan at the front end of the duct. It is rated for 650 CFM, and without any duct there, you will get essentially that. But suppose you put this fan in a duct. Moving air will see friction against the walls of the duct, slowing down the air flow. The back pressure reduces the flow, so you really don't get a full 650 CFM out the end of the duct. Of course, bends in the duct hurt even more, but you said it was a straight duct.
Now, suppose you add a second fan at the tail end of the duct. This is sucking air out at a nominal 650 CFM rate too. But in the end, all it does is give you something closer to the full 650 rating for the two fans. You don't get 1300 CFM, you get roughly 650 CFM.
With two fans in parallel, both feeding into the duck (or both pulling air out) if they are not limited by air flow through a duct that is big enough to handle one fan, then you might get more air through in theory. But the problem is, now you have higher velocity air. The link that gregmac provided tells us that pressure losses are proportional to the square of velocity. If you could double the velocity (and in order to double the air flow, you must double the velocity of the air as it moves through the duct) then you would quadruple those frictional losses.
It gets worse. As is pointed out in gregmac's link, with two fans right next to each other, the air flow into those fans will interfere with each other. So you really wont achieve the full rating for those fans if they are set side by side.
How about the 800 CFM fans? If they are too big for the size duct you have, then you again will be limited by the back pressure. You may end up with only a net of 650 CFM anyway.
It seems to make most sense to have a second duct. Run two ducts in parallel, both adequately sized for the fans they are fed by. Yes, I know this may not be an option.
It is difficult to properly cool two spaces with vastly different heat loads and only one thermostat.
Assuming typical computers and monitors and two users, you probably have about a kilowatt of heating or 3400 BTUs per hour.
Rather than run ducts, have you considered a dedicated cooling unit for your office? You can go with a small window unit. Even the smallest one is 5000 BTU and only uses about 500 watts when running. That's just a few pennies per hour and even that is offset by the reduction in load on the main unit. With the thermostat in the same room, its load response will be much better and the temperature extremes you have been suffering with will be eliminated. Plus you can turn it off when no one is there and save energy that way.
Lately my thinking is that ducts are basically parasites and efficiency goes up when the heater/cooler is in the same room as the load. In the past, the equipment cost was lower with a shared unit but operating costs (energy) have gone up a lot (and will continue to go up) and that could change the equation on selecting shared or dedicated equipment. Comfort will always be superior with dedicated equipment.
Best Answer
I don't see a problem with that, other than the existing duct doesn't look wide enough to fit both. You'll have to get/build a reducer that goes from something slightly wider (that can support both circular ducts coming out of it) to the existing square duct size. Most metal shops can build this sort of thing for you - I had a couple 45-degree elbows built to re-route some ducts in my basement, it cost me around $50 I think.
The main thing is to keep the number of elbows to a minimum. In fact, in your case it may work better as it looks like you'll reduce the number of elbows by one.
Another option would be to extend the main duct, and split the two round ducts off at the end. Not sure how long those are now, but the box stores carry a few regular square duct sizes you can buy and assemble/install yourself.