It was always a great honor when I was a student to be invited to a professor's house. It was never 1 on 1 - usually at the end of the semester, whoever was my current slavemaster (I worked through college as a student research or teaching assistant pretty much every semester) the proffie would invite us to their home. Some were very stiff occasions, but others were memorable evenings of great music (a fellow student on the piano and proffie's wife) and even better discussions.So now that I'm a proffie, I invite my slaves for Christmas cookies or a summer cookout or a beach volleyball tournament or whatever. They enjoy it, I enjoy it. And I must say that I am still very good friends with a student of 20 years ago (bicycle tours and watching soccer on TV) or the occasional beer on the town with former thesis students I took a liking to. Yes, they are friends. No, I don't go to bed with them. That's what you have a spouse for.
I'm with Katie - I have good friends among (former) students. But that doesn't mean that the rest of you squalid teachers-just-to-earn-a-buckers have to do the same.