I'm a member of the EFF, and its activities are great, but understandably have slightly more focus on USA-related matters. I say "but" just because I don't live in the USA.
Are there similar foundations that operate and focus/prioritize more on an international level? I don't manage to find any, but probably I'm doing the wrong kind of search.
I don't think so, not if you are looking for the same type of content. Applied privacy is very dependent on the laws of the specific country, this is why most organizations focus on one jurisdiction.
You do have organizations like Freedom of the press and Amnesty International that are international, but their main focus is not privacy and when they cover privacy it's more the ideology of privacy.