According to the Centers for Disease Control and Prevention (CDC), individuals diagnosed with strep throat or scarlet fever should remain at home until at least 24 hours after starting appropriate antibiotic treatment and after being fever-free without the use of fever-reducing medications. 
CDC
 This guideline helps reduce the risk of spreading the infection to others.
Regarding the need for a doctor's note, it varies by school or workplace policy. Some institutions may require a note to confirm that the individual is fit to return, while others may not. It's advisable to check with the specific institution to understand their requirements.