I think that a six-plex, while it would have some advantages (privacy being one of them) would not work in the same way as the OP's post.
First, you'd be buying six of every amenity: TVs, leather couches, pool tables, X-boxes, Netflix, Gamefly, housekeeping time, etc. This is a lot more initial outlay than the OP's situation.
Second, you'd have less discretion over who your roommates were. Actual landlords may want to chime in here, but it's my understanding that if you are the owner of a home, and you are choosing a roommate to live with you, then you have more legal discretion in excluding potential roommates (for example, if you're a woman you can refuse to rent to a man; if you're an old grouch you can refuse to rent to someone with kids, etc.) If you own the sixplex, this will be harder.
Finally, the rent situation will be different. In the sixplex, if you wanted to get your apartment paid for by the rent on the others, you'd have to charge about 20% more than comparable places with a similar mortage renting all six units.
In the right house, I think the OPs situation could be fantastic. For example, a large multi-level house with rooms with attached baths, or a split-level with a couple of bedrooms, a bathroom and a kitchen on both levels.
When I think about my roommate scenarios, a lot of the unpleasantness came because I really didn't have much control over where I was -- I needed to live with others to make it financially, but I didn't have any say in the others due to the rental situation. As the owner of the house, I'd have a lot more say, and my rules would be the main rules.
I think I could live with that, given the right setup!