People who calm you. The safest natural births are those where the mother can remain powerful, relaxed, and feels loved. If anyone in the room makes a laboring woman feel uptight for any reason, that person should leave. Family who makes comments like, "Why are you having another baby?" probably need to wait until after birth to see Mom. The same goes for those who will be so excited or talkative that the environment becomes nerve-racking. Ask any veterinarian, and they'll tell you that an animal in labor should not be disturbed or introduced to strangers. The corresponding stress hormones counteract a safe, pleasant birth. It's also important to consider how these people will react in the event something that constitutes an emergency happens. Will they remain quietly supportive, or will they have a meltdown? That will be the last thing you need.
People who don't want to rescue you. The deep relaxation that needs to happen in a natural birth can look a lot like someone in too much pain. We just aren't used to seeing someone swaying and moaning with eyes closed. Make sure everyone in the room understands that this is normal, even necessary. It's distracting during contractions for people to make faces or wring their hands. It undermines a laboring woman's confidence. This article comparing labor to a marathon explains this concept.
People who can focus on YOU! Your labor should not be the social event of the year. People who want to be invited only because so-and-so gets to be there probably will be thinking more of themselves than you. Someone being pouty and getting their feelings hurt should NOT be taken into consideration. If they can't understand that you are in charge of your birth environment, then they probably won't be considerate at the birth. You want folks who won't complain about how tired or bored they are, because I can guarantee, you'll have more to complain about. The tone in the room should be supportive and joyful.
People who know what you want. It is essential that anyone coming to your birth understand your desires well in advance of the baby's arrival. That way, they can remind you of your wishes when it gets hard; and they won't be surprised at what happens. Questions about the safety of a homebirth or the hospital's rules should be discussed ahead of time, not during labor. Assigning jobs ahead of time can help everyone feel included and helpful.