Bitters are a collection of bitter herbs but the precise type of herbs and proportions can vary between vendors and recipes. They are all intended to do what you described, increase salivation, gastric juices, peristalsis and biliary activity. Some claim to do more.
In terms of stimulating the biliary system there are three types of herbs. A cholagogue herb increases the bile flow from by gallbladder via contraction where a choleretic increase the flow of bile from the liver. Also; less precisely cholekinetic meaning increased bile flow.
Therefore choleretic herbs will increase bile flow for those without a gallbladder. Milk thistle, dandelion, artichoke and turmeric are firm safe favourites.