Your sentence is what is known as an ambiguous sentence. Taken on its own, the sentence
He is an English teacher.
can either mean
- a person who teaches English.
- a teacher from England.
Both interpretations carry equal weight - as a standalone sentence there is no way of telling which meaning is correct. One can argue that when spoken, inflection and sentence stress can allow you to differentiate between which meaning of the sentence is inferred.
He is an English teacher. (he is a person who teaches English.)
He is an English teacher. (he is a teacher from England.)
In many cases I agree that this can be true, however I don't believe it can be treated as a blanket rule, particularly if the speaker is speaking in a neutral tone.
However, the key phrase above (in my opinion) is 'taken on its own'. Rarely do we encounter these sentences as stand-alone sentences in real life - they are usually part of a much larger conversation, and as the saying goes, context is key. In real world situations we interpret meaning as part of a bigger picture - for example, if I were in the school staff room and was being introduced to the faculty, I can't think of any situation where 'John is an English teacher' would refer to anything other than the subject that John taught. The context of the conversation as a whole has led me to that conclusion, rather then the five words of that particular sentence.
In some ways, these examples are somewhat contrived, more to highlight the possibility of ambiguity, than to reflect any genuine real world confusion. Consider the following example:
I saw a man on a hill with a telescope.
- There’s a man on a hill, and I’m watching him with my telescope.
- There’s a man on a hill, who I’m seeing, and he has a telescope.
- There’s a man, and he’s on a hill that also has a telescope on it.
- I’m on a hill, and I saw a man using a telescope.
- There’s a man on a hill, and I’m sawing him with a telescope.
In real life we would work out the likelihood of what the speaker intended, based on the conversation as a whole.
All that being said, one of the comments in the original post hit the nail on the head. A native speaker would more than likely make the distinction by restructuring his or her sentence.
- He teaches English.
- He's (a teacher) from England.