No. When you design your invitation, there's a host field you control. You can keep what we propose (whoever you named as host) or write whatever you want. If you leave the host field empty, no host name appears to guests.
Pro tip: some hosts add a name and phone number so guests know who to contact with questions. The host name on the invitation is guest-facing and separate from shower host access, which determines who can manage your shower on Babylist.
People listed on the invitation as guests don't need host access or a Babylist account.