They don't break down if that's what you mean. They must be mechanically removed from the environment.
Most experts (which is to say people who make a living studying the stuff, rather than the lawyers who make a percentage of every dollar they can sue out of somebody over the stuff) agree that as far as "popcorn ceilings" and asbestos-containing floor tiles and insulation go, the proper thing for the homeowner to do is simply paint over it. This seals in the fibers completely, eliminating any possible exposure to the airborne fibers.
However, lets face it -- there is one good reason to get rid of those popcorn ceilings. They are ugly as sin (who the heck ever thought that looked good?). If you choose to remove the material, the following approach should completely protect you from any residual fibers after the job is done:
Use a good quality face mask with air filter -- this will cost maybe $20 at the hardware store. Even just a standard cheap filter mask ($1 style) will reduce the levels of inhaled fibers by 90% or more if used properly, but a good quality sealed one with a filter cannister will take out essentially 100% of any fibers.
Open the windows, and set up a fan to blow out through the downwind windows -- this will suck clean air in through the upwind windows, pick up any loose fibers, carry them out the downwind windows, and they will disperse outdoors in low enough concentrations as to not pose a threat to anyone.
Use water to dampen the material, this will keep it from forming loose fibers as well as make it easier to remove from the ceiling. A garden sprayer filled with warm water works best.
Keep the waste material dampenend and in closed bags once it has been scraped from the ceiling.
When done, dust thouroughly and vacuum the carpets and mop the floor, then throw out the vacuum cleaner bag and the duster and mop used.
BTW, don't freak out too much about asbestos -- the primary risk is just to those who are CHRONICALLY exposed to high levels of loose floating fibers of it, such as people who work with the stuff every day (insulation, demolition, concrete, etc workers in industries where measurable asbestos exposure happens every day).
Keep it in perspective:
Note that the risk of contracting a deadly cancer from casual asbestos exposure (the type that occurs in the home environment) is about 1/12000th the risk of dying from cancer due to smoking.
Also though it may be dry and overly technical, here is another great reference on the subject:
Note that this study puts the risk of contracting mesothelioma or other forms of asbestos-related cancers due to casual exposure this way: "Low-level exposure, as encountered in public buildings, probably does not represent any additional health hazard beyond what is incurred breathing outdoor air."