Chin implant: is it common not to use a screw fixture?

This is very common. You will find supporters of screw fixation of chin implants. IMO, this is unnecessary and every extra piece of hardware that we use during surgery can create potential issues/ complications down the road. Modern, anatomic type implants are long and fit into a very precise pocket along the jaw. As a result the risk of rotation and malposition of these implants is much lower than the older "cap" type implants. I hope this information is helpful for you.Stephen Weber MD, FACSDenver Facial Plastic Surgeon

The pocket is usually made under a thin but strong membrane called the periosteum with the pocket made to the exact size. The pocket then shrinks down around the implant to hold if firmly in place. A  a screw is not usually used for silicone implants (the most common type) but is used for others such as PTFE, Gortex, etc. 

This is a common practice.  I use three absorbable sutures to hold a silastic chin implant in position.  I am not sure there is ever a reason to use screw fixation, but it would not be unreasonable. If the pocket is dissected correctly and the implant 'fits' the pocket, then it will all work out well