Separation anxiety is a normal developmental stage we see in children. It starts at about 6 months of age and lasts till approximately 2 years of age. The peak of the anxiety oftentimes occurs between a year to a year and a half. It can be very distressing, and it manifests itself as an attachment oftentimes to a single parent and great distress on the part of the child when that parent leaves.
The best way to handle this is to be open in discussing it with your child. Sneaking away and leaving without your child knowing that you're leaving, or telling your child you're going to the other room when in fact you're leaving the home, might solve the issue in the short term but can be distressing in the long term because it results in your child not predicting what times you really are going to leave and what times you will be available. It's helpful for children with separation anxiety to talk about it early, long before the time to depart happens. An example: "Susie, I need to go to work tomorrow morning. Grandma's going to come over and take care of you. She'll give you breakfast and lunch and put you down for your nap, and by the time it's dinnertime, I'll be back home".
Another strategy that you can use to help with transitions is to have the caregiver come over early to allow there to be some time for your child to warm up to their arrival and to accept them as a new person to care for them while you're gone. For example, have your babysitter come over 30 minutes early to allow there to be a time for your child to interact with them and interact with you till they feel comfortable enough for their care to be assumed by the babysitter. Allow the time to be spent for those 30 minutes together in a room with the babysitter initiating some favorite activities of your child to speed the warm-up process along.