If someone scans a QR Code, do I receive their phone number?

No... you do not automatically receive the phone number of the person that scanned the QR Code.

You do however get information on the device and model used to scan the code. This site, http://en.wikipedia.org/wiki/List_of_user_agents_for_mobile_phones, may be of some interest to you.

In addition, you would also have information on the date and time that the code was scanned, as well as the IP address that the code was scanned from. With QReate & Track you can then click on an individual IP address and see a map of where the code was scanned geographically. (Please note: the location data associated to an IP Address is not always 100% accurate in regards to where the actual scan takes place. It may vary based on where a phone is registered or via which tower it's connecting to).

Of course, if you are looking to build your mobile database, you may want to direct people from the QR Code to a landing page that lets people enter their phone number and opt-in for future communications.