Personally, I would buy screened. I love being outdoors without the bugs. If you are unhappy, enclose it later. We may stain the concrete or add deck tiles. We will definitely be adding gutters to help keep it dry. We may also add a roll down shade to just one side where the sun hits the lanai in the afternoons.
Like another poster, we have a good-sized front porch as well - perfect for breakfast.
Outside is more fun. It's a personal decision - the sun only hits our lanai from one side part of the day and we have a fence, so there are no privacy concerns.