Who does the responsibility of product safety rest with?

The agencies that regulate product safety will vary with the product type - and the distribution thereof. If a product does not enter into interstate commerce, it is generally the state that is responsible. If a product is involved with interstate commerce (definitions of that could vary). then the safety of that product will depend upon the product type. Meat generally comes under USDA (there are exceptions). Foods other that meat (exceptions exist), dietary supplements, drugs, medical devices, biologics, tobacco products, cosmetics, and vet drugs come under FDA. The safety of consumer products (hoverboards, cribs, child safety seats, etc.) falls to the Consumer Product Safety Commission.