Very little, and it’s mostly in the form of light shining through the door.
“Radiation” just means anything projected through space. It includes potentially harmful neutron, gamma, and x-ray radiation, but it also includes sunlight, new ideas, and the petals of flowers.
The word “radiation” does not mean “magic death cooties,” and just because a device emits radiation doesn’t mean it can hurt you without falling off a counter and hitting your foot.
All electrical devices emit “radiation.” That’s because any oscillating movement of electrons—say like those in the wiring in your house—produces electromagnetic radiation.
This is not the sort of radiation that falls from the sky after a nuclear war or the type that bounces around inside a nuclear reactor or even the kind produced by traces of radioactive potassium in your bones—remnant of the stars that were here long before the Earth.
No, electromagnetic radiation comes in a range of frequencies or energy levels, from gamma rays at one end of the spectrum all the way down to radio at the other.
Gamma rays and x-rays, at the high end of the spectrum, can harm living things because photons of gamma ray or x-ray light are strong enough to break chemical bonds—and life is made out of chemistry.
Below X-rays on the spectrum is the so called “ultraviolet light.” UV light is right above the visible light part of the spectrum, and is right at the crossover point when electromagnetic radiation stops having the ability to break chemical bonds. The upper end of the UV band (called UV-C) can damage living things, and is used in hospital sterilizers for that reason. The rest of the UV band (UV-B and UV-A) can’t break chemical bonds, but can push them over the edge if they were about to break anyway, and so can damage living things in some cases. UV-B and UV-A is the light that gives you a tan and increase the risk of cancer—but it’s also important to normal vitamin D and cholesterol metabolism.
Below this, visible light is the middle part of the spectrum that we are evolved to perceive. We see in visible light frequencies because those are the frequencies that tend to be reflected off objects — higher and lower frequencies tend to pass right through solid objects, and so are less useful for looking around for dinner and harder to detect (because they pass through the detector too).
Visible light is the only kind of radiation that a normally operating microwave oven produces in any significant amount; it shines out through the door so you can see when your oatmeal boils over.
Radio and microwaves — at the low end of the spectrum — cannot break chemical bonds. The only way radio can hurt you is if you are not paying attention while listing to the top-40 hits on your daily run and fall down an open manhole. The only way microwaves can hurt you is if you break into a locked radar facility, climb up on the transmitter, and use it as your own personal microwave oven—and even that’s not very likely.
Microwaves (and radio) interact only weakly with matter. Microwave ovens actually required clever design even in order to heat up you burrito. If you were to defeat all the safeties and run a microwave oven with the door missing (don’t do that) the microwaves would mostly fly out through the room (radiate) and off down the street to interfere with the neighbor’s cordless phone.
To stop this from happening, microwave ovens are little microwave reflective boxes that bounce the microwaves back and forth and back and forth and back and forth and back and forth and back and forth and back and forth and back and forth and back and forth and back and forth and back and forth and back and forth and back and forth and back and forth and back and forth and back and forth and back and forth and back and forth and back and forth and back and forth and back and forth and until they finally soak into the freaking burrito already — adding a little heat.
When a few quintillion photons of microwave energy are absorbed by a burrito, it starts to warm up. If it warms up enough, it’ll cook and melt the cheese nice and soft the way you like it. If it warms up too much for too long, it’ll grow tough, dry out, burn, and maybe even catch fire—none of which has anything to do with radiation or how the heat was applied.
The only way you can tell this is happening is to look through the door. The door to a microwave oven is a Faraday shield—a conductive grid made with spaces far smaller than the 4.8 inch wavelength of the microwave “radiation” that prevents it from passing through.
Some leakage does occur. In the United States, federal law (21 CFR 1030.10) limits the leakage to 5 milliwatts per square centimeter at approximately 2 inches from the oven surface. Since EM radiation spreads out through space, this means the dosage would be no more than 0.05 milliwatts per square centimeter 20 inches from the oven, and of course that assume it’s operating.
Thus, for the poorest designed, crappiest maintained, most powerful microwave oven, there’s about 3,500 times more energy inside the oven than there is 2 inches from the door—and the energy inside the oven actually wouldn’t hurt you beyond warming you up. Microwaves are, in fact, used in certain medical applications to safely warm living tissue.