Main locations on the human body which cockroach bites are Face (eyelashes), Mouth, Hand, fingernail but cockroaches do not transmit diseases/infection directly from a bite. Instead, they contaminate surfaces (fomites) and foods which later turn into the source of the disease.

People with asthma are most likely to experience an asthma attack when exposed to the allergens from a cockroach; hence is advisable to always clean the environment and dispose garbage properly.

Thanks for the information @Kehinde