In actual fact, this is not a major problem at all in the professional sphere. As a native English speaker teacher, you will normally be expected to conduct all of your exchanges with staff and students in English. The majority of professional correspondence and contracts will be in English (although there may often be local language counterpart documents). Admin staff at an English language centre almost always speak good English too.
In class, the students are there for an English language immersion experience. If you start speaking their first language to them in class then you are likely to be jeopardising the quality of their learning. In addition, most of your students will already have had English classes and therefore be able to communicate with you throughout the class in English only. Indeed, allowing them to use their first language can often be a sign of a teacher who has poor classroom management skills.
Outside of work, basic local language skills are certainly highly beneficial. It will make shopping, travelling and socialising far, far easier although you may find it surprising just how many local people want to practise their English language skills with you anyway! If you haven’t learnt any basic local language skills when you arrive you do not need to worry too much. The best way to learn is to throw yourself into situations where you inevitably will learn (sometimes much faster than in a traditional classroom context) such as shopping for fruit and vegetables at the local market. Think of it as a fun task to strike up a conversation using the local language at least twice a day and try to learn a minimum of five new vocabulary words each day too. Within a few days you will be getting the hang of it.
If you haven’t started teaching abroad yet, have a look online for books and audio cds that may be worth buying to give you some local language ability before you arrive. This will improve your confidence when interacting in your new environment. Once you have arrived, see if you can watch Western films with local language subtitles to aid your learning. Note that this only really helps beginners if you are in a country where the Roman alphabet is used such as Indonesia, Malaysia, plus most of Europe and Latin America.
Some teaching contracts include free or discount price local language sessions in order to help you to settle in better. If you are offered these be sure to go along. You will learn invaluable skills and make good friends along the way. It is also possible to offer your English teaching skills in exchange for local language skills from a new local friend on a one-to-one or small group basis. Everyone benefits from this and you will learn a great deal more about the local culture this way.
It is also a great idea to do some research into expatriate communities in the city you will be living in. There are often running clubs (such as the Hash House Harriers) or more general social events for other expatriates including English teachers such as yourself. In the beginning, you will probably want to seek occasional advice from those who have lived in the different culture longer than you have. Further down the line, you will probably enjoy sharing anecdotes about life in both cultures with others who can closely relate to you. The people you meet in these contexts will usually be more than happy to give your their own tips and tricks about improving local communication skills and understanding more about your fascinating new host culture.