Accepting that the Haftara should be read by someone competent, this is not always the case in many shuls. During leyening from the Torah, which is usually done by an experienced, regular ‘professional, it is important to correct any mistake, however, as the Haftara is not read from a claf (which means that anyone reading it for himself in a Chumash is Yotzei anyway) should one be stringent in correcting mistakes, at the risk, perhaps, of vershaming the reader? What sources would be used in making this decision?
It is the custom in some places for each individual to to read the haftarah to himself asnd not listen to the public reader. It is not clear that we can fulfill our obligation from the public reader if there is no klaf. Certainly one should not correct him if there is any chance of embarassing him. BTW, in the Torah reading very often people make many corrections. It is only obligatory to do so if the missaid word changes the meaning of the verse. It is proper to have someone knowledgeble in Hebrew to stand near the reader and correct him quietly, and ask all others to refrain from yelling out corrections, both to maintain decorum in the shul and to avoid embarrasssment.