As all individuals are different in nature, as are their needs, interests, goals, and most importantly in this field: learning styles.
Finding the best fit for your child, as well as the teacher, makes the year go by with less bumps, hiccups, and meltdowns of any sort.
For the Student: your child is unique in their own ways, and the teacher must understand this. Some children work well with teachers who automatically box all kids into the same clump -therefore, teaching them in their set ways...aka, how they teach every other child, because your child is flexible and listens to what adults tell them to do. However, in my experience, I've seen children, and teenagers flourish in their musicality because of tailored lessons. After a few weeks or a couple of months in working together, the teacher begins to notice strengths, areas to work on, interests/passion,and work ethic for each individual student. Tailoring lessons to your child allows for maximum learning with the student being self-motivated and engaged, leading to *hopefully* consistent daily practice each week.
Example 1: I've taught one teenager for about 3 years now, this year she just started Grade 8 in school. After a few weeks of lessons together when she first started, her piano lessons are unique to her: we begin with Technique (RCM based), she is assigned to compose a piano piece or work on a song of her choice -by ear!, then 1-3 classical pieces (RCM). Note that this may not work with everyone! She is a very responsible teen, practices at a minimum 4 days every week, and follows/listens to directions quite well.
Example 2: little ones who cannot sit still, do not really listen well, and like to do whatever they please. These students are a little more challenging to create a solid plan for, most times, lessons are re-tailored each week. Over time though, they settle in after getting to know you as their instructor, and then you can set in a flexible lesson structure.
Example 3: gifted students! These are a rarity, but they do exist, and I've had the wonderful experience teaching several gifted students! Also a challenge to set a solid structure for, as they are very curious, their minds wander with questions and stories to tell, and getting them to focus is tough -but achievable over time.
(I could sit here all week and list all the students I've taught over 5+ years! Each one is different in their own way)
For the Teacher: depending on the teacher's personality and what parents want out of the lessons. Some parents prefer the strict, traditional style of teaching, whereas others like a modern thought. I myself am more on the lenient/nice side, understanding when students say they had lots of homework -HOWEVER! if it becomes the excuse each and every week, I know it's not honesty and then I have a talk with them about the importance of practice, and also inform their parent/guardians. It's one thing to be kind and understanding, but it's not okay for students to come up with excuses not to practice. It doesn't do anyone any good.
NOTE: It's OKAY to try different teachers when first starting out, it's important to meet the teacher and understand their style, what they can do for your child, and how they will work with them to achieve goals.
Here are some links to useful articles regarding Choosing the Right Fit for your Musical Lessons:
Piano Teachers Federation: Choosing the Right Teacher
Kawai US: How to Find the Right Private Piano Teacher
Royal Conservatory of Music: Finding A Teacher (directory)