I have been a nanny for the better part of the last 18 years and I am a mom of a almost one year old boy myself. Some babies are just going to be that way no matter what and they will out grow it. I had one little girl that loved me until her mom left and then she would scream until the mom returned for about 6 months.
In the mean time here some things you can do for comfort.
NEVER sneak out when the child is not looking! This will only increase the anxiety and clingy-ness.
ALWAYS explain that you are leaving and that you will return soon. This will build trust. They may not happily wave goodbye but they will understand. Don't linger or draw it out. Just gather your things, give a cuddle, say goodbye and then leave. Sometimes the screaming will be just for your "benefit" and will calm down when you are not there to hear it.
Have your child spend some time around other care providers when you are present. Hire a babysitter to play with your child while you get some house work done. You may not actually get a lot done but your child can build a relationship with this person without being resentful at them because when they come over you leave. Even if you are able to do it have the sitter change the diaper, refill the sippy cup etc. It will help them get used to having other persons meet their needs.
If you have a specific routine, favorite song, game, order of doing things when you are home make sure you communicate the importance of these to the sitter. The familiar and routine will be comforting to your little one.
Hope this was helpful.
B.