I'll echo mmouse. After 9 cruises, we've had about every configuration possible....2 of us at a table for 4, 2 at a table for 8 with 4 other people (2 groups of 2), 2 of us with a family of 3.... One time, we were asked if we would mind joining a group of about 15 (this was as a special favor to our serving team, and is not a routine thing...the other family was also asked, and it ended up being a really fun thing).
If you ask that the 7 of you be seated together with no one else, you will be. Also, if there is a problem at the first night, you can ask to be moved.
We've never had a problem when we've had dining companions. I think that
DCL makes a real effort to seat "like" families together. WHen we were seated with 2 other twosomes, each had a girl within a couple years of my daughter's age. One was travelling with a mom, one with a grandma. When we were with a family of 3, their daughter was within 2 years younger than mine.
I can't imagine that there would be a problem keeping your group "alone together." Just ask now (if you've made reservations).