Private space on the internet would be anything on the LAN side of your Router, public space would be anything on the WAN side. You may own a domain and have it hosted at a datacenter, but that does not mean you have the right to conduct scans against that domain, as all that you own is the domain. You do not own the equipment it is hosted upon. Nor do you own the equipment inbetween you and your domain.
Here's an example:
My friends and I own a few domains, we have them hosted on our own server that we built ourselves. The server is hosted at an ISP that we pay for rackspace. While we may have the right to scan our box because it is physically ours, we could still get in trouble because we would be using the equipment in between us and that box. If the ISP has an IDS system in place, they could detect the scans and report the attempts.
Most AUP/TOS are worded in such a way that you may not conduct recon's while using their service.