There are number of private schools (colleges) in GTA. I will list a very few below;
1. Trios
2. Access Business
3. Everest
4. Canadian Business
5. Maple Leaf
6. LSBF
Private colleges might not be best options for International students comparing the community colleges. Students applying to Canada must clearly understand, they are applying to Canada not UK. In UK, you have colleges which offer UK university degree but Canadian colleges does not give you degree from a Canadian university (they do not have they liberty, as far as I have seen here)
Realistically, how many International students will NOT like to work here? Almost every student will like to work atleast 20 hours a week to cover their weekly expenses associated with their living. In my opinion, Private institutions are much suitable options for immigrants or citizen who are seeking a job switch or on EI, as these colleges gets funding from government to support such employees. Also you can certainly not compare the size of any community college with any of these private colleges.
Also the most imperative of all for me is your diploma from a community college may be convertible to a degree into a university, as most of those colleges (you name it) have articulation agreements with the university, whereas you possibly will not get credit for the same if you do from a private college. After visiting almost all colleges in Ontario, I hardly see any private college listed on college degree transfer arrangement? I may be wrong but do you know any? If you find one, let me know I can find this for you from the relevant person in the university.
Regards
YM