What level are you hoping to reach in your target language?
Depends on the language. In general an intermediate level or even just tourist level is sufficient for me.
What kind of teacher/tutor/language partner could help you achieve it?
For a language I'd like to be fluent in: highly proficient in the target language, needs to know the ins and outs and a stickler for grammar. Should be strict with me. For Chinese: needs to have really good pronunciation.
For a language I'm intermediate in: good listener, interested in a lot of things, intelligent, there should be chemistry between us
For a language I'm a beginner with: should know how to teach me without resorting to English, has to be patient
Are you confident that you can choose the right person yourself?
Mostly yes. In most cases I was quite happy with the teachers I chose. I even found the best teacher of my life on italki.
- if you're looking for a native speaker: can you tell whether the person you found is a native speaker or not?
Not on a beginner level. I'm afraid I can only tell for English. Even though my Japanese and Chinese are pretty solid, I think I could still be fooled.
- if you're looking for someone who understands grammar: can you tell if the person you found has the relevant knowledge?
Yes, because I'll ask very specific questions and if they can't answer them and I find out later, that the rules are pretty much straightforward, I'llbe disappointed And also on beginner level, I'm able to see if a teacher is bad with spelling.
Think of people with needs/expectations entirely different to yours (i.e. much higher or lower).
Could they be taught by the same person as you?
Maybe, maybe not. I'm really into grammar, spelling, etymology, linguistics. Other people might be bored by that. But good teachers can cater to different students.