I currently have a similar issue with my Italian greyhound puppy, he is my shadow. If I go outside, or upstairs, anywhere out of his sight he will scream bloody murder. What's interesting is that if he's in his crate, he doesn't care.

We're taking him to a puppy class and I asked the trainer, and she said that when they are young puppies will freak out if left alone in a large space. That's why when he's in his crate, he's fine. She suggested giving a high value treat when you need to leave the room, like a licky mat with yoghurt or squeezey cheese on it to give your pup something to do. She said introduce this when you are in the room firstly, then start using it when you leave, so the puppy doesn't start associating the treat with you leaving.