Does a chin implant sit under the depressor anguli oris muscle?

Chin implants are placed along the lower border of the jaw directly on the bone and below any orbicularis or depressor anguli musculature of the lower lip.

yes. Chin implant lays under all muscles of facial expression. They are either placed over or under periosteum. 

Hi, yes, the silastic chin is typically placed under the periosteum which is beneath the muscle(s).  I have performed many facial shaping procedures, including Chin Augmentation with dermal fillers or silastic chin implants, for over 30 years.  When the chin is weak, this creates an imbalance making the nose appear larger, the mid face top heavy, the lower face looks short, de-emphasizes the lips and allows early formation of a "double chin".  
 Proper placement of a silastic chin implant adds forward projection to the chin thereby creating harmony and balance to the lower face.  Using the same incision, liposuction can be performed to reduce the fat and further shape the neck.  Excess skin, from below the chin, can also be removed through the same incision.  
 I have found that placement of a silastic chin implant, through a small curved incision under the chin (also allows excess skin removal) to be very safe, quick, highly effective and far less invasive than a sliding genioplasty (requires extensive tissue dissection, bone cuts and placement of metal screws and plates to secure the cut segments of bone).  I perform chin implant surgery in 30 minutes or less, often using a local anesthetic alone.   When consulting with surgeons perhaps ask not only is your chin weak but do they have experience performing chin augmentation?  The attached video shows how I determine whether the chin is weak.  The selection of the proper size, shape and thickness as well as the proper placement of the silastic chin implant are all "key" elements in creating a naturally looking chin.  If too large of an implant is used in a woman, the chin can look relatively masculine.  That's where experience, performing chin implant surgery, comes into play.Hope this helps.