I'm giving two weeks notice next week. I work in a cube environment, my boss does not have a separate office.
Should I just do it at his desk (in front of everyone) or request privacy?
This is a no brainier: you do it somewhere where the discussion won't be overheard. This gives your boss the ability to manage the news of your departure as they want, rather than it coming out via gossip and rumour.
If your are a skilled professional / salary: private meeting.
If you are hourly / high turnover position: private is preferred, but may not be practical or necessary. His cube is probably fine.