Qualifications needed to teach math/statistics at US university
A friend with a B.S. in Mathematics and an M.S. in Statistics would like to teach math/statistics at a US university which is not research oriented as a full-time faculty member.


I have some questions.  First, do you mean a specific university, or will any university do?  Second, do you really mean "university," or could it be a "four-year college" or a community college?  Third, by full-time do you really just mean full-time, or do you mean tenure-track?  

A few data points:


In my department (mathematics, UGA) we have faculty with master's degrees who are teaching full-time, but they are not tenure-track.  Some of these positions are "permanent" by any reasonable standard -- e.g. some of these faculty have been there for multiple decades.  A friend of mine just got hired into one of these positions, so I know a bit about them and I don't think they're going away anytime soon.  In fact my friend has a bachelor's in math and a master's in statistics.
At my PhD program's Visitation Day on Friday, I met with a woman who is a faculty member in the math department at a four-year college relatively nearby.  She has a master's degree and has been teaching there for about five years. 
In some regions of the country (e.g. New York City) even community colleges tend to hire people with PhDs.  But there is a lot of variation.  You can read e.g. here for more details.  My understanding is that even in the community college scene having a doctorate makes a difference in your job titles and responsibilities.  



  However, my friend is not interested in statistics research, so she's thinking instead about trying for a PhD in Math Education.


The "so" confused me.  This makes as much sense to me as saying "physics" or "computer science" in place of "math education."  In other words, these are all academic fields that are closely connected to mathematics but distinct from it.  So far as I know (and I do know about the math ed PhD program at my university, although I also know it's one of the best in the US and therefore is probably not representative of all departments) being a math ed PhD student is much like being any other PhD student: you are being trained in research and implicitly/explicitly groomed for an academic research career (though many graduates will end up elsewhere).  

So getting a PHD in math ed is an excellent idea if your friend is deeply interested in math ed research.  Well, and also if she can find funding: in my experience, one of the main differences between math and math ed PhD programs is that the former are much better funded than the latter.  But since you didn't mention any prior training in that area, I'm a little concerned that you are viewing this as some kind of degree for people who are interested in teaching rather than research.  That might be true for certain people and/or certain programs, but it's not true for all.  

The idea of someone joining a PhD program in field X because they don't like field Y makes me very nervous.  I would certainly encourage your friend to learn as much as she can about post-secondary academic jobs available to people with master's.  A combination of math and statistics sounds rather valuable to me, and it may well be that your friend is employable now.