If you registered a domain name, it is yours for as long as you continue to renew it.
When you put the "privacy" feature on, it's harder to see if they registered it in someone elses name or yours since the "whois" details only show you the "privacy company"
I don't recommend using the Privacy feature on a domain name for anybody that is running a business. That's information that the public can use to help "research" your business to make sure you are a reputable person to do business with.
I know that I, for one, tend not to do business with businesses that have "privacy" registrations. It takes away another way of contacting them (email and phone) if something goes wrong with the purchase. It also makes me think they have something to hide.
But yes, the domains are yours and all you have to do is login to godaddy to "manage" them and manage the DNS to point to whichever host you want to use.
It's possible that you'll have problems if you use GoDaddy as a host. It's possible that you'll have problems if you use any web host as host
If you do a forum search for godaddy, you'll see lots of opinions on godaddy as a host and as a place to register a domain. I've seen lots of good and "so so" posts about them as a web host.
Are my domain names mine??? 
