Glycerin is routinely derived from animal fat, the vast majority of which is either from non-kosher animals, non-kosher slaughtered animals, and even if not, it is then forbidden fats - whatever is easily obtained as a byproduct of meat production. Historically this was the common source (initially coming from candle production, and then from soap production), but as bio-diesel and other processes that involve vegetable oil have increased, it is of lesser importance as a source.
Kosher Glycerin is generally derived from vegetable oil instead. It can also be chemically synthesized from petroleum.
According to a glycerin product manager in Cargill quoted here in 2012, 30% of Glycerin in the United States comes from non-Kosher animal sources, and 70% from vegetable based sources. That wouldn't address any equipment sharing concerns. (Synthetic glycerin from petroleum is not generally used in food).
The process from animal fat is a straight forward hydrolysis (separating chemical bonds with water), which requires lye (which can be gotten by boiling ash from hard wood) and salt.
Rav Moshe Finstein writes (Y.D. 1:62 - the subject is the Kashrus of blended whiskies) with regards to Glycerin that it has no taste and is added to whisky for non-taste reasons (to make it smoother to swallow).
The website kashrut.org puts a big emphasis on Rav Moshe in arguing to allow Glycerin (I don't know if this is the only teshuva where Rav Moshe discusses it).
However, both assertions of fact made in that Teshuva are incorrect. Glycerin has a sweet taste and is used to enhance the natural Glycerin that is a byproduct of production as it naturally occurs during firmentation. Yes, it makes it smoother to swallow - in part because it removes bitterness, being a sweetening agent. (You can see here about its natural presence in fermentation (and a mention of its sweetness) and find many recipes online which use Glycerin as a sweetener. It is also used in dog food, sometimes as a sweetener).
(On the subject of the Teshuva, the Glycerin still may not pose a Kashrus problem for blended whisky, but just not for the reasons stated, rather because they don't add it by government law, or if they do it is in very small amounts. However moonshiners still use it).
Almost every major* American Kashrus agency will list in an article on their website glycerin as a Kosher problem. Rabbi Belsky requires mouthwash to have a hechsher specifically because of the beneficial presence of glycerin in it (it creates the fresh feeling) - although others will permit it because it is not intended to be swallowed.
One of the big sticking points for Rabbi Gefen in his original certification of Coca Cola in 1935 was the Glycerin even though it was in amounts that were less than one part in 60.
*As defined by professor Timothy Lytton in his book Kosher. In fact, he uses it as an example of the type of ingredient that must be certified up and down the stream of food production.