Mispronouncing Words

Shalom! 

Thank you for your question. 

There are a number of interesting sources that are relevant to your question. 

The Mishnah (Megillah 24b) raises the issue regarding leading prayers and birkas kohanim. Someone from Haifa and Beit Shean may not lead services because they confuse the alef and ayin sounds and the heih and ches. Rashi offers two reasons why they may not lead: 1) By using an ayin sound rather than an alef in birkas kohanim, they change the blessing into a curse. Additionally, 2) their mispronounced prayer is defective. 

The Shulchan Aruch Harav, OC128:48 quotes two reasons for disqualifying such people from leading prayers: 1. they distract the public with their improper pronunciations, 2. their prayers are defective. 

The practical difference between these two views is significant. According to the former, when the public is used to the mispronunciation, such as an established member of the community, there is no problem with him leading services. According to the latter, only in a place where most people pronounce Hebrew that way is it allowed. 

There is an interesting responsum of the Radbaz (1:399). He was asked whether Greek Jews could recite birkas kohanim in Egypt, where their pronunciations sounded foreign and incorrect. He replied that while their pronunciation of the blessings was sufficient for Greece, where all the Jews spoke Hebrew that way, they could not do so in Egypt. The Magen Avraham (53:15) and the Mishna Berura 53:37 agree, 

Circling back to your question, not every mispronounced word of the Ba'al Tefillah needs to be corrected. The rule is that only words whose omission would prevent one from being Yotsei, and even then, only when the mispronunciation changes the meaning of the word.

This can be quite complicated, and thefore it is the responsibility of the Rav or one designated by him, to correct the Chazzan.