We just got a new computer last year. Our house is very small and the living room and dining room are really one big room. The computer is in the dining room so there is really no privacy there. The old computer we moved to the basement playroom, but it has no internet access, so they can really only use it for the word processor and to play CD-ROM games.
We were also thinking about getting a laptop which my older son will be able to use for school (but it will really be my computer) with the stipulation that there is to be no internet access for him while he is in his room. If he wants to access the internet he will have to do so in the living/dining room area.
My Daughter has had her own computer since she was 1 1/2 years old. We keep it in her playroom. My Husband has it set up so she can only access her websites and that no popups will come up. I also keep an eye and ear out. My Husband and I have our own laptops.