Yes, there are, and are readily available.
https://www.chpa.org/OTCRegulation.aspx

Regulation of OTC Medicines

The distribution of medicine in the U.S. is based on a two-class system: prescription and nonprescription medicines. A nonprescription, or over-the-counter (OTC) medicine, is one that the U.S. Food and Drug Administration (FDA) has found to be safe and effective for direct consumer use based on the label instructions and warnings. FDA requires a wider margin of safety for OTC medicines than for prescription drugs. FDA has subjected all OTC medicines to safety and effectiveness reviews since the 1970s.
https://www.chpa.org/OTCRegulation.aspx

The prior respondents discussed the FDA regulations for over the counter drugs. These products must either adhere to established monographs that detail the formulations and allowable claims or they must go through the drug approval process if they have novel ingredients or novel claims for established ingredients.

There are also regulations regarding non-drug personal care products such as cosmetics. The distinction between drug and non-drug cosmetics lies in the intended use. Products that are marketed to be therapeutic or to prevent damage (even sun damage from sunburn) are considered drugs even if they do not contain active drug ingredients. Please see this FDA guidance document for more details: https://www.fda.gov/Cosmetics/GuidanceRegulation/LawsRegulations/ucm074201.htm

In the case of OTC toothpastes, there is the Caries Final Monograph, on the FDA website, that clearly specifies a range of tests that are acceptable for confirming the efficacy of fluoride-containing toothpastes.

In Europe we have this prescription / non prescription dichotomy as well. However, what we call OTC is not really over the counter as in most countries, the pharmacist will give the product that is behind the counter, but without a prescription It is assumed that the pharmacist will give advices and check that there is no contra-indications, or that an adult product is not going to a kid. In most Europe countries, all drugs are behind the counter and cosmetics and devices are over the counter. However, in the industry we use the term OTC to refer to non-prescription drugs.

As a drug is a drug, an OTC drugs need to be developed according to the same standards than prescription drugs. What makes a prescription drug is often the safety, the sides effects that may require a medical control, the therapeutic index, ...