I just had to comment because a random person walking by could get the impression that we offer legal advice, which we don’t.
I agree it’s an interesting question, and I was under the impression that private school teachers (as opposed to buxiban teachers) also need teaching licenses like their public school counterparts, but I don’t recall where I may have read that.
Edit:
Employment Service Act Art. 46 Par. 1 Subpar. 3
http://law.moj.gov.tw/ENG/LawClass/LawAll.aspx?PCode=N0090001
3.Teacher at the following schools:
三、下列學校教師：
(1)Teacher at a public or registered private college/university or school established especially for foreign residents.
（一）公立或經立案之私立大專以上校院或外國僑民學校之教師。
(2)Approved teacher teaching course(s) on foreign language(s) at a public or registered private primary or high school.
（二）公立或已立案之私立高級中等以下學校之合格外國語文課程教師。
(3)Teacher teaching course(s) at a public or registered private experimental high school’s bilingual department or at bilingual school(s) .
（三）公立或已立案私立實驗高級中等學校雙語部或雙語學校之學科教師
Qualifications and Criteria Standards for foreigners undertaking the jobs specified under Article 46.1.1 to 46.1.6 of the Employment Service Act
http://law.moj.gov.tw/Eng/LawClass/LawAll.aspx?PCode=N0090031
Chapter 4 Teaching Work
Article 40
A foreigner to be hired to engage in teaching as set forth in Article 46.1.3(1) shall obtain a teacher certificate pursuant to the Regulations Governing the Screening of Qualification on Teachers of Junior Colleges and Higher Levels. Alternatively, he or she shall be reviewed by the eligible faculty of the school or college to determine his or her qualifications. An exception is the foreigner who is to work as a teacher at a school meant for foreign residents, or as a foreign language teacher at a foreign language center affiliated to a college or above.
外國人受聘僱從事本法第四十六條第一項第三款第一目規定之教師工作，
應依專科以上學校教師資格審定辦法取得教師證書，或經學校審議通過其
任教資格。但外國僑民學校教師或於專科以上學校附設外國語文中心擔任
外國語文教師，不在此限。
NB: The “Regulations” mentioned above (專科以上學校教師資格審定辦法) are filed in English under the title Accreditation Regulations Governing Teacher Qualifications at Institutions of Higher Education. Regulations Governing Accreditation of Teacher Qualifications at Junior Colleges and Institutions of Higher Education - Article Content - Laws & Regulations Database of The Republic of China (Taiwan)
Back to Art. 40:
The foreigner who is to work as a foreign language teacher at a foreign language center affiliated to a college or above, as set forth in the preceding Paragraph, shall obtain a degree from a domestic or foreign university, or an independent college recognized by the central governing authorities. In addition, the language courses taught by the foreigner shall also be in the official language of the nationality specified in the passport of the foreigner, and approved by the central governing authorities.
前項專科以上學校附設外國語文中心擔任外國語文教師，應取得中央目的
事業主管機關採認之國內外大學或獨立學院學位，且教授之語文課程應為
外籍教師護照國籍之官方語言，並經中央目的事業主管機關同意。
Article 41
Foreigners to be hired for teaching work specified in Article 46.1.3.2 & 46.1.3.3 of this Act should acquire formal degrees in colleges or universities either in this country or overseas that are recognized by the authority concerned at the central government level as well as the teaching qualification for the subjects they will be teaching.
外國人受聘僱從事本法第四十六條第一項第三款第二目及第三目之教師工
作，應取得中央目的事業主管機關採認之國內外大學或獨立學院學位，且
取得擬任課程合格教師或任教資格。
Article 41-1
The quota for foreigners to be hired for teaching work specified in Article 46.1.3.2 of the Act shall be no more than the outcome after the total number of school classes authorized by the authority concerned multiplies by the number of foreign language studies sessions per class per week defined in the course syllabus, plus the total number of language studies sessions per class per week in the school exceeding that defined in the course syllabus and then divided by the maximum weekly working hours of each subject teacher authorized by the authority concerned, except the senior high schools which are approved to organize the second foreign language studies session.
雇主聘僱外國人從事本法第四十六條第一項第三款第二目教師工作之員額
數，不得超過目的事業主管機關核定之學校總班級數乘以課程綱要規範每
班每週之外國語文課程節數，加計超出課程綱要規範之全校各班每週語文
課程總節數後，除以目的事業主管機關核定科任教師每週最高上課時數所
得之數額。但高級中等學校獲准開辦第二外國語文課程者，不在此限。
I don’t feel like digesting it right now, so feel free. 