Multilingual search problem
I do not as yet have significant experience with internationalisation in drupal.

Having said that, if you would like greater control over your search indexes I would definitely look at using search api instead of the default drupal search.

As well as allowing you to pick which fields are indexed, it also abstracts your search index so that you can take advantage of Apache solr etc. In your case, at least to start, I would suggest you stick to the simple database backend for search api.

The more flexible approach of search api brings additional functionality via other modules. For your use case I would look at Search API Entity Translation, which


  generates separate search index items for all/selected available
  languages for each translatable entity


It looks like this relies on entity translation rather than internationalisation, so make sure you survey any potential implications for translation of your content first.