If a jet engine is bolted to the equator, does the Earth speed up?
It's the latter. Look at the system earth + engine + atmosphere. Conservation of angular momentum must hold for the whole system (assume no gases leave the atmosphere due to the engine, which is a fair assumption).
Briefly, the Earth's rotation rate will change; and the net rotation rate of the atmosphere will change in the opposite way. But very soon, friction between the air and the ground will ensure that the air and ground have no net rotation difference, which means that the Earth's rotation will return to the original rate.
Suppose you have a large torus floating freely in space. The torus is sealed and full of liquid. Inside the torus is a propeller (or screw, if you prefer). I believe your question is reasonably equivalent to:
If the propeller in the torus starts spinning, will the torus experience angular acceleration?
Hopefully it is obvious that the propeller will accelerate the fluid in one direction, and that this will impart angular velocity to the torus in the other, thus causing it to spin [faster]. At the same time, once the propeller stops turning, you should be able to figure out why the higher angular velocity won't last: friction will eventually reduce any difference in angular velocities between the fluid and the torus to 0.
Obviously, the earth and atmosphere are not a torus or water, and the actual dynamics of the air is much more complicated. However, this should make clear why the most common answer of: "It spins faster briefly, then slows back down again" is essentially correct.
On the other hand, if you want a more permanent increase, just leave the engine(s) running. You could create a permanent "jet stream" at the level of your engine (much more effective if you have many engines placed around the equator).