Although not essential, it is recommended.
Bacteria is only thought to survive on your toothbrush for around 24 hours in moist conditions and even less in an air dry space due to the fact that they generally die when exposed to oxygen, however when you are sick it can linger.
This can lead to reinfection or passing it on to other members of the household so it is also best not to store your toothbrushes anywhere that they are touching.
The infections that we are mostly at risk of transferring are group A Streptococcus or Strep throat and cold sores.
The Strep bacteria can live up to 3 days on a rinsed toothbrush and 15 days on a non-rinsed toothbrush.
Dentists recommend that you change your toothbrush after a taking prescribed antibiotics for 48 hours if you have Step throat.
If you do not want to buy an new toothbrush you can clean it by boiling, using hydrogen peroxide, vinegar or leaving it in mouthwash for 20 minutes.