Seems like this idea is actually being pursued by some one who knows what they are doing.
Pandemic Ventilator Project
They use a bellow design. I wonder how this is superior to a fan based system? The DIY Powered Respirator described here might be modified.
An estimate of the amount of energy required to breath is given here. Which back of the envelope is less then 20 watts. The fan in the DIY respirator is 13 watts so my sums are probably wrong.
So can a pc fan or a squirrel cage blower be made to operate for 3 seconds pumping 500 ml of oxygen into the lungs and then to allow this air to escape in 2 seconds? A microcontroller could be used to time this.
An alternative design is some sort of Iron lung. This gets around many of the nasty infection problems a putting a tube down your throat could lead to. However according to here
"This style of ventilator would be of no use in a flu outbreak, in as much as flu usually floods the lungs with fluids, and you have no way to remove the fluids."
So maybe homemade iron lungs are not the way to go?
Another thing that would be needed are sensors to
1. Signal when the machine failed to work correctly.
2. Signal when the patients gas exhalation or pressure were not normal.
Why might a DIY respirator be necessary? Some of the mathematics of ventilator shortage are on the Pandemic Ventilator Project site. And the cold numbers on survival rates without a ventilator are here.