I don't think anyone outside of you can determine whether some or all parts of you are integrated -- united -- or not, though I think many therapists do state or assume integration has happened. For any alter beyond a fragment, I would think you would have a sense whether they are part of your united consciousness or not. Admittedly, we've had zero integrations here so I'm no expert. I get this from reading of bios of persons with DID and from individual reports here. We have experienced blends where there is sometimes an increased awareness of being the same individual. From what I've read, with actual integration, there is a clarity that you are both alter A and alter B. There's also normally an ability to recall memories belonging to all integrated alters, though that doesn't necessarily happen immediately.
I would think that if you are not really integrated, thinking you are would get in the way of moving toward actual integration. I've dealt with so many therapists who have been inadequate to incompetent regarding DID and, sadly, I know that's not unusual.