cookiemunstahh comments

My background = have Viet, half white and live in a neighborhood (Los Angeles) with many immigrants... went to school with immigrant children from all sorts of generations. I also minored in Linguistics at UCLA.

In my opinion, from what I've learned in linguistics combined with personal experience, the best way for your child to be "very" bilingual is ONLY talk with your child in a language not spoken by the general public in the country where you live. My friends whose parents did not do this usually cannot read or write in that language and have only limited speaking skills (can carry on a conversation about normal household things or the weather but not an intellectual conversation). I notice the ones with best bilingual skills had parents who didn't know any of the country's language and relied on their kids for translation (you of course don't have to rely on them for your own communication). Those people tend to be extremely fluent in both languages.

My friend, amusingly, is Taiwanese and she was born in Chile. She moved to the US when she was a teen. Her parents and step-father only spoke Taiwanese or Mandarin. As a result she is extremely fluent in Chilean Spanish, English (her English is amazing... usually teen learners have a bit of accent but she has no accent), Taiwanese Chinese and Mandarin Chinese. She learned German at UCLA and is now extremely fluent in German. The only problem she has is she can't write very well in Chinese and she is a beginner at French... none of which she was super exposed to.

So, in sum, if you flag and speak some of the language of the country you are in (say, Japanese)... their Japanese will be stronger than their English, for example. They simply will not have been exposed to complex conversation in different languages.

International schools are iffy--I think you still have an accent because you reasons stated above... except it would be reverse... their Japanese would be weak. (I moved a lot and also went to a French International School briefly as well.)