In essence, toothpastes work by mechanical surface abrasion, so they scrub the plaque and remains of food from your teeth. They can even be ranked by their abrasive effect measured by standarized methods (e.g. Relative Dentin Abrasivity), althought this might be misleading, so do not follow such rankings blindly:
RDA values are not intended and should not be used as prediction tool of dental abrasion, since it does not reproduce the complex multifactiofial nature of the toothbrushing abrasion process clinically.
Additionaly to their main purpose, toothpastes may supply fluoride or other substances which can affect tooth remineralization (self-repair), reduce ability of the plaque bacteria to produce acid, or maybe raise pH acidity level in the mouth.
There is no definition of "clean tooth" or "dirty tooth", but there were attempts to measure how many times a day teeth should be brushed so they will be "clean enough" to maintain oral health. Quote from "Tooth brushing and oral health: how frequently and when should tooth brushing be performed?" (2005):
(...) there is consensus in the literature that (meticulous) tooth brushing once per day is sufficient to maintain oral health and to prevent caries and periodontal diseases. Tooth brushing is also regarded as an important vehicle for application of anti-caries agents, such as fluorides. However, most patients are not able to achieve sufficient plaque removal by performing oral hygiene measures at home. Therefore, tooth brushing twice daily is recommended by most of the dentists in order to improve plaque control. This rule is followed by most of the patients taking care for their oral health and has shown to be effective in maintenance of oral health in numerous studies.
It should be noted that tooth brushing may not be sufficent to maintain oral health under some conditions like improper diet (highly acidic food or drinks), improper drinking technique, regular attending of public pools (so called "swimmer's erosion") or maybe even using old toothbrush (supposedly you can disinfect them with vinegar but I've never done that). Here it is advisable to use an alkaline mouthwash as those bad effects (and probably many others) are connected to low pH. When mouth pH levels are low tooth enamel begins to demineralize (even without bacterias). Additionaly, low pH is a good environment for some species of bacteria which produce even more acid.
It seems there might be a may to actually see 'dirty' tooth after all. You might want to use plaque disclosing products which contain dye reacting with plaque. There seems to be wide variety of such products: tablets, mouthwashes, flosses. You can find many videos on YouTube showing how to properly use the tablets. After using such product you can see where the plaque is, how efficent your toothbrushing (or any other method of plaque removal) was, what places were left etc.