Combustion of either propane or natural gas can create dangerous conditions for indoor air quality without adequate ventilation.
Propane is a hydrocarbon that ideally consists of length-3 carbon chains (C3H8 also known as C3). However, cooking propane is never 100% pure, and contamination from soluble hydrocarbons from C10 up to C40 is likely.
Natural gas is a mix of hydrocarbons primarily composed of methane (CH4) but also longer hydrocarbons and alkanes. In Canada, natural gas must be at least 90% methane to be legal for sale. Some common impurities in natural gas are sulfur compounds and mercury.
Under idealized circumstances (100% pure methane or propane and 100% complete burn) the only combustion products are heat, water, and CO2. However, ideal circumstances are never achieved in the real world.
This study shows that normal combustion of propane is likely to produce carbon monoxide (CO) because of less-than-perfect combustion efficiency and particulate matter (PM2.5) because of impurities and contaminants.
There is no substantial safety difference between propane and natural gas when used for indoor cooking. I strongly recommend electric (either resistance or induction) heat for safe indoor cooking.