Do I need separate insurance if I'm traveling internationally?

Yes. Kaiser doesn't cover health care costs you incur in other countries. Even in the states, they will only emergency room visits at non-Kaiser faciities and then only if you call and have it approved before you go. Otherwise, within the US, you can visit a clinic and pay out of pocket and file a claim later.

Internationally, there are insurance plans for expats, students, and business and leisure travelers. You can pick the amount of coverage you want, copays, etc. You're usually covered for one trip or if you're a frequent traveler, a certain number of trips a year. These plans will even include "evacuation" - getting you home as quickly as possible if the health care system where you are visiting is substandard, delivering your body back to the States if necessary and travel for a family member to come be with you while you are hospitalized. Just search online for "travel medical insurance" for the various companies that offer it.