calorie in-take and out-put

If you are asking if you can eat less than 1200 a day because you burn 1200 in exercise. I would suggest against. MFP always shows you that if you eat less than 1200 you are not getting enough calories. Your body would go into starvation mode causing your metabolism to slow.

You should be eating at least 1200. If you are burning 1200 additional calories you could up your caloric intake to 1400 and still have a difference of 1000.

You really need to NET 1200, which means eat your exercise calories. MFP intends for you to have a deficit if you do no exercise, so working out that much you need to replace the calories so your deficit doesn't become too large and start to burn muscle mass. If you have larger fat reserves, you can usually get away with having a larger deficit, but if you're closer to your goal weight it's really important not to get to the point where you're losing too much muscle. Muscle helps burn calories!

So I have a question that bridges both the Food and Nutrition topics and the Fitness topics

I have recently upped my working out to two hours a day when I can, (and as close to it on days I can't). My daily in-take according to MFP is 1200 calories, but the two hours I spend on the elliptical burn almost all of the 1200.

Is it okay to not eat a net of 1200? As long as I am hitting at least 1200 without exercise?

Let me guess a couple things.
MFP shows your BMR around 1400.
MFP gave you maintenance calories of 1800 based on your selection of activity level sedentary.
You selected 2lbs weight loss per week.
MFP set your goal to 1200.

But if you do the math for 2lbs / week (7000 calories), that means a deficit of 1000 a day from your maintenance calories.

So that would have put you at 800 daily goal, but MFP has some safeguards, so left it at 1200. So that isn't even a 2lb / wk goal anymore. Sadly it has no such safeguard for suggesting eating below your BMR.

Now lets assume that estimate of calorie burn is correct (may be close actually) at 1000 calories for 2 hrs.

So you eat 1200 calories.
You burn 1000 calories.
Your body for all other functions has 200 calories to use now.

Your basic energy needs for a healthy metabolism was estimated (we'll assume close) at 1400.

Do you see a problem here? Please don't do this to yourself.

Your body at healthy metabolism can now barely get the energy it needs for the basic functions of life. Not even talking about getting out of bed yet to do anything else. The energy needed to digest food, repair muscle, ect. If this is truly done every single day and diet matches. It will lower to respond to what you are feeding, or rather, not feeding it.

In case BMR is not understood - http://en.wikipedia.org/wiki/Basal_metabolic_rate
energy in this state is sufficient only for the functioning of the vital organs, the heart, lungs, nervous system, kidneys, liver, intestine, sex organs, muscles, and skin.