Derms definitely do face lifts- from what I understand the techniques were developed by dermatologists as an offshoot of their flaps and grafts on the face in the last century. Derms also developed liposuction. Both usually require a fellowship post-residency. I've only seen derm surgeons do eyelid lifts (blephs) and "mini-facelifts" b/c most cosmetic surgeons out of any field don't seem to work in academics.
One benefit to having a dermatologist do either of the above procedures is that they tend to use local or regional instead of general anesthesia (vs., say, plastics) resulting in faster recovery times (ie procedure on Friday, back to work Monday) and fewer AEs (the only recent deaths from these procedures in the area I trained were due to complications of anesthesia).
Would I recommend a dermatologist or a plastic surgeon?
Like anything, the experience in the particular procedure (ie, how many they've done, how long they've been doing it, who they trained with, etc) would probably be a major deciding factor.
Anyone else have info on this?