From the first day Sean came to live with us (a rescue adopted at the end of February) he wanted to be with me ALL the time. He would cry and bark when I left his sight. Now he's okay if he's with my husband. A week after he came, I kept leaving the room to let him know I'd be back.
He's funny, though, as others have said when he has to know where I am. When I go into the bathroom he looks to see where I am, happy that I'm not far away. His real reason for looking, I think, is so he can jump up on my chair, his favorite. Funny guy! As soon as I return he looks at me, realizing he must get off, so he gets up and actually follows my direction for him to get off so I can sit down.
My first Yorkie came to live with me when he was older. (He had been with my parents.) He was always by my side when I was home.
My other two had each other for most of their lives so they weren't as dependent about being with me.
As said by others, depends on the dog. Most important is the love that we show them. Sean and I became VERY CLOSE from the beginning.