Most people of course are not at either extreme but somewhere in the middle. This is likely good for society!
However, it's through exploring the extremes that we can find our own tendencies. Since we're not having a face-to-face conversation right now... be honest with yourself: What are you looking to do? And where does this put you on the spectrum? Sure, your goals likely will change over time in one direction or the other, but you need to be able to evaluate which teacher in the area might be a good fit for you right now. (Or whether it's worth driving out of the area or studying with someone over Skype.)
Before even considering a trial lesson, you should do some research. This is the part most people skip, and it's probably the most important part of the entire equation. Even if you lean more toward the left side of the spectrum, where this is mainly a fun activity with few serious long-term goals, you'll at least want to make sure that a prospective teacher has a minimum proficiency on your instrument, a history of teaching, accepts students in your (or your child's) age range, at your (their) current level of ability, and you'll want to find out about other details including pricing, where their studio is located, policies, etc. This should all be easy enough to find on their website. Should all of that look good, it could be this is all you need to be ready to contact them about a trial lesson.
The farther toward the right and toward the more serious side of the spectrum you lean however, the more additional research there is for you to do. How qualified is this person to teach what I'm interested in learning? Do go ahead and read their bio and teaching philosophy to get a sense of their education, professional experience, and ideas. Find them at YouTube, listen to recordings. Ask any good musicians you know for advice/recommendations. Only at this point might it make sense to take a trial lesson with someone.
While one-off trial lessons are not a great time to find out about their qualifications (that should be done before), they are good for getting an assessment of where you currently are and come up with a plan going forward. Trial lessons are a good way to give you a sense of whether you can get along with the prospective teacher or not. You want to make sure that both of you are on the same page as to what you're looking to accomplish--- and that you simply like being in the same room together for an hour at a time! Longer trials such as a month or three months are obviously better to see how you will work together, but if you don't have that luxury, one will give you a general idea.
In lieu of research, some people like to schedule a week of trial lessons to "audition" multiple local teachers, which we'd have to advise against. To start with, unless you happen to already be an advanced musician yourself, you likely don't know enough to assess who is the best teacher for you/your child from one trial lesson alone. Friendliness is great, but it shouldn't be mistaken for ability either in performance and/or teaching. Wherever you fall on the spectrum, you'll need both of these to *some* degree. Secondly, not all teachers will agree to be 'auditioned' in this way. If they sense a parent is letting their young child choose, and the child is shopping around for the most fun, easiest experience (i.e. no one will make them practice!), some will simply not agree to the audition, and you may be missing out on a real opportunity.
Anyone can say they are a private violin/viola teacher. Let me repeat that: There is a wide range of people who will offer to give you lessons in exchange for money, and there is no regulating agency to oversee them. There is no sort of federal, state, or local licensing procedure. If you wouldn't hire a plumber without a license and recommendation, then you shouldn't hire a violin teacher without the right credentials etc. either!
There are many highly skilled teachers out there simply with differing competencies.... and you should do the research to find one who is right for you. We specialize in teaching adult students, and we have taught quite a few students younger than that as well. However, we're good at assessing and are very up front if we don't think we're the best fit for your needs. As you are putting your (or your child's) education in the hands of someone who will be a strong influence on you (or them) musically and perhaps generally as a mentor as well, we believe in not wasting your time if we're not the right fit and will happily refer you to someone else.
In the age of the internet, research is easier than ever and the best tool you have to do the work to find the right teacher for your needs.