No, they call them "Independent" schools, because being free from both government control and funding has positive connotations apparently.
The teacher's and the school may refer the school as an Independent, but the pupils all call them private schools. At least everyone I know who's been to one does.
Now, technically I went to a private school, and I was on the staff for a short while (as an IT tech.) but I still call it a private school. (Awesome, I just checked the school's wikipedia page, they've now added a list of people I've never heard of as "notable alumni" brilliant!)
[edit: aww nuts! Just checked the Wife's old school's wikipedia page, and I recognise EVERYONE on their "notable alumni" list.]