@Alice

I can see that there are people out there who have downvoted you, but your edited version completely contradicts your original post, in which you categorically stated that unless each and every error was corrected then no teaching was taking place and the teacher wasn't doing their job properly. If this is your teaching method that's fine, but I find your edited version rather disingenuous.

I type out the mistakes the students make during a conversation then go over it with them at the end. :)

If it's going over vocabulary I correct them if they make a mistake. 

In my personal opinion, there's no "best method" about the correction of mistakes. It depends on the student's goal. Many people here pointed out that if they get corrected all the time, they will lack confidence to continue speaking. I don't. When I am a student, if I hire a teacher, I want to be corrected. Of course, if my mistake is related to a grammar point that I never touched before, either the teacher should teach me later, after I finish my thoughts or not correct me at all at that particular grammar topic. But if is a new teacher, I try to let them know what grammar topics I've studied so far so they know what kind of mistakes I'd like to get feedback on. Same goes for pronunciation, intonation and etc.

The difference about me between many people on italki who hire teachers is that I don't pay for "language exchange partner kind of lesson" where the main focus is "pure conversation". Money is not easy for me, so if I'm paying, I want to receive feedback according to my level and needs. And when I hire teachers, it is mostly because I want to learn and understand new things and concepts of the language that I have difficulty or that I am lazy at doing by myself. If I just want to talk, I try to get language partners which I am a bit lucky with. 

As a teacher, I go for whatever my student wants. Each person has a unique learning style and I try my best to adapt to that. Also, I would say that I have the gift to understand what kind of student I am dealing with, I know how to approach a student that is open to receiving constructive feedback or those who are more sensitive to them. Experience!. When I get students that are influenced by the mafia of the famous polyglots (Benny Lewis for ex:) and they say "I want to learn by speaking from day one, I hate grammar, don't even mention grammar topics to me, don't correct me and bla bla", I still do teach them grammar.I  still correct them and they don't even notice it. 

Well I've done some TEFL and in my opinion teachers shouldn't interrupt a student every time they make a mistake because this interrupts the flow of their speech and becomes VERY annoying! Instead, the teacher can write down the mistakes quietly while the student is talking and then explain the mistakes to them at the end of the class. It's not necessary to tell somebody about every single tiny mistake they made, because that can really affect their confidence. It's best to just tell them about the most important mistakes. 

Another way to do it is for the teacher to respond to what the student is saying, but saying it correctly, like this:

Student: Yesterday I go to the park.

Teacher: Oh, how interesting. So yesterday you went to the park! 

This method also involves interrupting so it's not a very good method in my opinion. However some people might prefer it that way and I think all private teachers who are doing one-to-one classes should always ask students which method they prefer. The class should be done the way the student prefers, not the way the teacher prefers. The student is the one who is paying for the class so they should be able to have it the way they want. 

Part2

Now compare what I would call good practice. The teacher/tutor allows me to speak freely and make all sorts of mistakes, only stopping the conversation if communication really breaks down. After a certain agreed length of time or after an activity we pause and review serious or minor errors depending on my level in the language. This is where Benny's 100 errors start to be corrected and real teaching happens. Any teacher worth their salt will have been noting down errors during the preceding conversation in the dialogue box/google doc or invisibly, and an experienced teacher will know what best to focus on.( Or might even suggest an area of difficulty to address in a subsequent lesson/ send some consolidation type exercises to work on between lessons)

Crucially the student then puts in time and effort to review corrections and work on any suggestions before the next session.

This method works for me as an absolute beginner in Russian and a C1 learner of Spanish but unfortunately I have also had the bad experiences ( see Part1) in all languages I've ever learned. It's the worst feeling!

I feel really fortunate and very happy to have met and worked with many excellent teachers and community tutors who either have an enlightened attitude to correction or who are willing to adapt some of their strategies to help.