The olive oil/citric acid " flushes only affect the gallbladder, not the liver. Ingesting any dietary fat stimulates a release of bile from the gallbladder to help with the absorption of the fats. Olive oil though is also rich in sterols. Sterols have a very high affinity for cholesterol and binds with the cholesterol in the intestines forming an insoluble complex that looks like gallstones. Therefore people assume they are passing gallstones, when in fact they did not come from the gallbladder at all. The cholesterol is found in the foods we consume and from the bile salts released from the gallbladder. What proof is there that these "stones" are formed in the intestines? To start with real gallstones sink due to their calcification. Most people report their "stones "float". This is because these "stones" are a mixture of cholesterol and sterols. Secondly, there is no way possible for many of these so-called stones to pass through the bile ducts if they were real gallstones. Some of the photos people have displayed of these so-called "stones" show "stones" from the size of a dime to a silver dollar. Even with the magnesium to dilate the bile ducts, there is no way possible for a real gallstone, even the size of a dime, to pass through the bile ducts. If gallstones were really present then they could lodge in to the bile ducts from the large amount of olive oil causing severe gallbladder contractions. If this occurs the person will end up in the operating room for emergency surgery. So these "flushes" should NEVER be done. So why do people feel better with these "flushes"? Simple, they are moving bile in the gallbladder preventing stagnation of the bile. But this can be done a lot safer with small amounts of oil with each meal, such as a salad with oil and vinegar dressing.

Bitters on the other hand actually flush the liver. When bitter tasting substances hit the back of the tongue they stimulate the bitter receptors on the tongue. This in turn stimulates the vagus nerve. In response there is an increase of stomach acid, bile (which prevents gallstones and helps to reduce formed stones), and increases pancreatic enzymes. This is why they are called "digestive" bitters. The vagus nerve stimulation though also stimulates the liver to flush and has a  slightly stimulant effect on the intestines to help expel the toxins being released. So it is a good idea to drink a lot of water throughout the day when using bitters to help dilute and flush the toxins. This is also why bitters are so effective in lowering cholesterol and balancing out hormones as both are regulated by the liver. 

Getting back to the gallbladder, the most common reasons for gallstone formation is a lack of bile and bile stasis. Bile contains a lot of lecithin, which is a fat emulsifier. So the lecithin in bile helps to keep the cholesterol in the gallbladder in solution. If the cholesterol levels exceed the saturation point of the bile the excess cholesterol precipitates out forming gallstones. Stagnation of the bile contributes to this as cholesterol will continue to enter the gallbladder. If it is not being carried out on a regular basis the bile will quickly saturate and excess cholesterol coming in to the gallbladder will quickly precipitate out.

Contributing factors? Lack of fiber and lecithin, both used in the formation of bile. Lack of intake of beneficial dietary fats leading to stagnation. And estrogen and progesterone. Estrogen increases cholesterol levels in the gallbladder as where progesterone relaxes the gallbladder leading to reduced contractions and stagnation. This is why women are more prone to gallstones than men.