Good questions and no they're not invasive, so no worries. Each day (and sometimes each interaction) the person acting as "me" is just one of the entities taking over, so effectively there's no central "me" at all if that makes sense. Today it's introverted family-man, but if we attend an event in the evening for example, the more social personality takes over and listens in on the conversations from family-man and others. Let's say the social one is currently active and sitting at the bar with friends having a great time, the family-man will say "we should text our wife and let her know we'll be home late" to which the social one may respond with "nah, we'll do it later" or "yeah that's a good idea, we'll do it now".

EDIT: to answer your last question about "equal access", we share thoughts and deliberate, but one can't exert control over the others nor forcefully take over. The active person at any given moment depends on the outside environment and what we decide is best for the situation, so it's kind of a democratic process; not sure if that can be defined as equal access or not.