Taking initiative can be really hard. Some are too shy to speak up and ask questions regarding problems they see. Some don’t want the added work, so they avoid speaking up. DO NOT DO THIS! I find people who notice things above most designers are the best on the team. They notice problems in the briefs, in solutions and in the companies image. By having such a trait you become vital in the effort to avoid false advertisements and save face for the company. This is especially important when working with a startup because they may not know the huge role design plays in the companies success, you have to be the one to show them.
If you’re confused as to how to develop this skill it starts in school. You must give critiques no matter what. Say something, ANYTHING. When someone presents solutions or problems learn to defend properly, argue properly and give options properly. These are the components to taking initiative.