There is a degree in teaching Chinese to foreign speakers. Lots of tutors will have this, don't settle for less.
I would also look for someone with experience teaching.
The CET4 is the College English Test level 4 (4000 words), CET6 (6000 words) which is really a bit like IELTS level 4-6. A person with CET is not an English major.
At the very least I would suggest a minimum a tutor had is TEM4 (8000 words), the college Test for English Majors. But all English Majors and those who will teach Mandarin as a foreign language must pass TEM8 (13000 words).
CET would not really prepare someone with the linguistic knowledge to teach and explain.
Neither CET nor TEM are teaching certs.
My experience of 3 types of study is as follows:
I have tried university study, but this did not suit me. All 4 skills (reading, writing, listening, speaking) taught.
Private tutor, 80/hr and I could only afford 2 hours twice per week. Tutor was not 100% available, no lesson planning and so no real structure. Lessons in public are were not ideal. Advantages, I could choose my own books, I did not have to learn to write (not 100% needed unless you are looking for qualifications) and one to one. Focussed on reading, listening, and speaking. Was also able to focus on a specific topic if needed, but it never really was. As a beginner you need functional language, individual vocab can be looked up.
I am now in a private school. This is a perfect fit for me. Cheap, regular and structured. It works out about 20 rmb/academic hour. I study 2 hours per day M-F which is still affordable. This is a small group of 5 if they all turn up and this really is not a problem. Almost the same as one to one, really it is. There is a planned course and the tutor is always there. This is a classroom with a whiteboard, and we have time to explore individual teaching needs when they occur. There is no great focus on writing, which helps. In everyday life, as long as you know PinYin to put into your computer/phone and can recognise which character you need to use, that is enough, and is what most young Chinese rely on.
my 2c.