I get up early, well before the kids. I've been out of routine for a while, sleeping in and taking some time off in the summer, so now when I get up early I have that queasy feeling kind of like jet lag and need to finish waking up before I can write legibly. I write in a notebook and transfer it to computer if I want to share it.
For sleeping, our family shares the big bedroom upstairs, which frees up the downstairs rooms. One is a guest bedroom, and one is what we call the "craft room" where the kids and I have our tables and desks for schooling and creating. Somehow, I have let all my "to do" stuff in there, bills and lists, and I have to try to keep them somewhat contained or they can really drain the inspiration and sidetrack me. I have a big table in there where I paint, and that is also where I write.
For a while, I kept a tin of cookies in the closet in there, to reward myself for just getting in there and sitting down to write. I ought to do that again!
I also keep an "idea notebook" for jotting down bits and pieces of things that have caught my interest during the day so when I do sit down I have something to write about immediately.
My dream space would look similar. Only it would end up such a mess, and I'd have more hours on the clock. I know what you mean about "their" space, eemamahob. I've had to really work to carve out mine. Over time, it's grown.
And yes, it does get easier as the kids get older! Much, much easier.