The usual term is make-up artist or makeup artist .
These may be professionals who work with actors in various productions like theater and movies, as well as in the modeling industry.
Here is more on this.
They may also be professionals who help "ordinary" people with their makeup, typically working in salons along with hair stylists.
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From MW: cosmetologist noun : a person licensed to provide cosmetic treatments to the hair, skin, and nails : one trained in cosmetology : beautician EDIT TO ADD: From the NY State Division of Licensing Services website, emphasis mine:
The practice of "cosmetology" means providing service to the hair, head, face, neck or scalp of a human being, including but not limited to shaving, trimming, and cutting the hair or beard either by hand or mechanical appliances and the application of antiseptics, powders, oils, clays, lotions or applying tonics to the hair, head, or scalp, and in addition includes providing, for a fee or any consideration or exchange, whether direct or indirect, services for the application of dyes, reactive chemicals, or other preparations to alter the color or to straighten, curl, or alter the structure of the hair of a human being.
It is the responsibility of licensees to understand the Appearance Enhancement Law.
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Also from MW we have cosmetician A person who is professionally trained in the use of cosmetics
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It depends on what kind of makeup she applies, and in what context.
Somebody who works with actors and actresses on the set of a movie or TV production, or with models on professional photoshoots, is referred to as a makeup artist.
Somebody who works in a beauty salon with individuals who will wear the makeup in their daily life is referred to as a beautician, cosmetologist or aesthetician .
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A person who works with makeup and specializes in it is a makeup artist .
A cosmetologist usually provides treatment to the skin, scalp and other parts of the body.
Some cosmetologists might do makeup as well, I think they also learn to do makeup when studying, but a cosmetologist usually has very little to do with makeup.
Search on Google for Hung Vanngo and Pat McGrath, whom are the two most important makeup artists
and I think your friend might be studying to become like them, and see how they're labeled/called.
You'll notice they're never called "cosmetologists".
Source: I'm a woman who is very much into makeup and skincare and goes to spas often.
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They are usually called makeup artists.
In India it's a common practice to call them a Makeup Man.
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I have a friend that studies  make-up  in College and she is already a professional that works with it.  What is the name of this professional?    For example, a person that works with architecture is an architect.
What is a person that works with make-up called?
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It depends on what kind of makeup she applies, and in what context.   Somebody who works with actors on a movie or TV set, or with models on photoshoots, is referred to as a makeup artist. However, somebody who works in a beauty salon is referred to as a  beautician, cosmetologist  or  aesthetician. It depends on the type and context of makeup one uses; either makeup artist or aesthetician
It depends on the type and context of makeup she uses.
Anyone who works in films or TV productions or in professional shooting models together with actors and actresses is called a makeup artist. 
Someone who works in a beauty salon, who is a beautician, cosmetologist or aesthetician, with individuals who wear make-up in their daily lives.