Hi, I moved to Spain 15 years ago with 2 small children, my daughter was 4.5 at the time. With no knowledge of spanish at all, they both went to spanish public school. I found it tough with regards to teaching writing and reading but we spoke english at home.
Both children are fluent in both languages, my daughter took a high level english exam which enables her to attend english university if she wants, but she opted for spanish uni - she is in Seville studying Biology and is the top student in her class.
The public schools here are excellent, the children are young enough to pick the language up quickly, especially if you praise and encourage them.
Secondary school can be hit and miss as they have so many classes - at that point I put mine into spanish private school, and they get in mainly on merit, not on how much money the parents have! Therefore standards are high.
Personally the education here was higher than the UK (we went back for 8 months and the only homework both kids got was painting and drawing whereas in Spain they had science and maths questions!!!) It is a very thorought system, the kids are constantly tested to ensure there are no gaps in the education and can be held back a year if they fail at 2 or 3 subjects which they have to retake.
I cannot tell you about the international ones as I have no idea, some are good, some arent so good - look for the results and speak to other parnts and their kids to see how well they think it is.
I can tell you my 2 wouldnot have had their education any other way and as fluent spanish speakers they had the option of a 3rd language. They took French in the public school and German in the private one, thereby giving them 4 languages.
Good luck in whatever you decide.