I believe in the value of a liberal education. A bachelors degree in computer science should prepare you for a lifetime of learning, no matter where you end up pulling a paycheck. You may decide that you want to work in a completely different field, but want to pursue independent game development on the side. You are much more broadly valuable with the university degree.
Another way to look at it: go with whatever is cheaper. I am not a professional game developer, but my understanding of the biz is that companies hire based on what you know and what you can do, not where you graduated. If you can develop something significant, no matter where you went to college, you’ve shown that you can work in the business. However, I would wager that the private school is much more expensive, and you’ll spend years of your life paying off debt instead of something more fun like buying a house or a nice car.
Note that my comments assume that your local university has a quality computer science program. Talk to their faculty and students. Find out what kinds of opportunities are to be had: can you get involved in independent studies to customize your education? Can you get involved in faculty research projects? Are courses taught by professors or by underpaid/overworked graduate students?
I hope that helps.