Most Taxidermists like myself , do not tan personally due to the amount of time and work required to tan 300 deers plus other animals, so we send it out to be professionally tanned, either a wet tanned or dry tanned.
This makes for a better and lasting mount.
Someone with experience such as Bill Yox or myself can probably spot a none tanned hide easily, while amateurs can't seem to tell the difference.

Shrinking around the eyes, muzzle, or mouth pulling open, is sign of dry preserve or just a pickled hide.
This is a very cheap way of doing deers, there are ways of reducing the shrinking by using a strong hide glue and plenty of pins to hold the eyes and nose and mouth in place until thoroughly dry.

Post a few pics and I will be able to tell you more.