You are quite right that you will get different answers. TESOL teachers usually see their role as humanely facilitating assimilation of learners to the national (English-dominant) society, and rarely know the language(s) of their students or feel any need to know it. TEFL teachers, almost by definition, usually know the language of their students, and have different perspectives on the goals of their instruction. Unfortunately, a contagious ideological view has spread abroad that the goal of language teaching is personal communication, with grammar instruction being rejected (partly because of the lack of training on the part of the teachers). The older model, for good or ill, was aimed not at interpersonal communicative skills, but at a larger knowledge of academic language, especially in reading and writing. One can guess that the majority of international Nobel Prize winners learned their English this way. Students taught following a communicative curriculum with little attention to grammar may enjoy their classes more, but rarely progress to academic competence (and most, unless they were very motivated from the outset, quickly forget most of what they have spent years studying). It is worth noting that focusing on reading is more efficient for learning, and removes some of the inadequacies sometimes felt by TEFL teachers.
As a rule, I always recommend that every TESOL teacher should put him/herself in the learner's seat and go through the experience of actually struggling with learning a different language -- preferably one that is grammatically and lexically very different from English -- at least once every five years (it takes about that amount of time to forget what the experience is like). I have learned more about what is most effective, and especially what NOT to do, from putting myself in the learner's seat this way than from any amount of reading or abstract exhortation.
As teachers of English as a foreign language we've been exposed to a variety of methods and methodologies that have populated the teaching of English throughout decades, some more concerned with grammar, others with communication. We have reached a point where all this struggle to find the best way of teaching a language has come to a dead end street because the post era method suggests an eclectic approach to teaching where teachers are allowed, better said invited, to seek those approaches and methods that best fit the needs of their particular students and a given context. As the antics would say, undoubtedly the truth lies somewhere in the middle: you cannot teach a language without addressing grammar, and teach grammar without using it in real conversation, yet generation of students (myself included) did a wonderful job in learning not one, but several foreign languages using the Grammar Translation Method, the Audio-lingual Method and a few others, so indeed it is at the teacher's latitude to select those aspects of grammar that can make a difference in students' conversational abilities.
One of the books that I would recommend, a classic in this respect is Betty Azar's "Fundamentals of the English Grammar" with its latest editions and its corespondents for different levels of proficiency.
Thank you for your answer, Adriana. While I agree wholeheartedly with your endorsement of an eclectic approach to language teaching , I think you may have misunderstood my question. What I would like to discuss on Research Gate is as follows: which grammatical features typical of spoken as opposed to written English (fronting and tailing, spoken hedges, spoken discourse markers, repetition and recasting, spoken ellipsis, spoken clausal structure etc) need to be taught in order to help students improve their production and understanding of conversational English?.Some recent grammar books, including the Cambridge Grammar of English by Carter and McCarthy and the Longman Grammar of Spoken and Written English by Biber et al cover a number of grammatical features which occur predominantly in the spoken language. However, very little of this has filtered through to the grammar content of textbooks and spoken grammar is rarely included in courses of conversational English.