Chin Implant Complications - What do I need to be aware of before getting the implant? Can they "slip" out of place?

When a solid silicone implant is used, which is the most frequently type used today, there are essentially 3 most common complications from placement of a chin implant (known as "Augmentation Mentoplasty"):
1.  Numbness (known as paresthesia) of the lip and/or chin.  There is a nerve that exits the mandible (jaw) on either side of the chin called the mental nerve.  When placing the implant through the tiny incision of the skin under the chin, the nerve cannot be visualized.  During the procedure the full thickness of the soft tissues are elevated off the bone of the jaw in creating a precise pocket to place the implant into.  As the pocket is created, these soft tissues are pulled and pushed, in which the nerve lies.  Nerves are extremely delicate, and can easily "fall asleep" temporarily from this indirect manipulation, without EVER touching the nerve itself.  The mental nerve is only a sensory nerve, not a motor nerve.  Thus, it is not uncommon to temporarily lose some sensation on one or both sides of the lip and chin.  Nerves of the head and neck, if not cut, take weeks to months to recover, and rarely over a year if not permanently damaged.
2.  Malposition i.e. change of position the implant can occur.  This is very rare if the pocket is made very precise to the shape and size of the implant, and is secured to the bone.  Firmly attaching it to bone can be done with a single bone screw placed through the implant and into the jaw.  I prefer using an absorbable stitch placed through the implant and the lining of the bone, called the periosteum.  The ultimate goal, no matter which technique is used, is to maintain the proper position of the implant as the body creates a capsule around the implant the following several months, which is what ultimately holds it in place for you lifetime.
3. Infection is extremely rare.  With proper sterile technique, and also placement through the skin instead of through the mouth, infection is still possible, but very unlikely.  If it does occur, it is noted within the first several weeks of surgery, not after.  Treatment requires removal of the implant and oral antibiotics.  A new implant can be placed after the infection is completely resolved, and with proper sterile technique, because it happened once does not mean you have an increased possibility with the second implant.
Hope you found this answer helpful.  All the best!

Complications can occur with any surgery but the specific ones for chin implant include the following: scarring, bleeding, implant infection, numbness, muscle injury, migration of implant, lip drooping, anesthesia risk.  Seeking a plastic surgeon who specializes in facial surgery would help mitigate these risks.