Nora, you can "breakfast" all day if you want to. Just walk 100 yards to your right to Helen's Creperie. Or to Graziella to your left from Horizon. And many others.
Horizon is a favourite of many. Sea view available - right on the beach. Or Marina - many positive posts about there and also on the beach. The sea front offers everything you ask.
Privacy is easily available, but it depends what you mean. Villa Linakis or Villa Theia are nearby. Or go with Olga at Elena Stella where we stayed all of September 2008. Or Tria Adelphia at the edge of the village.
Send me a Personal Message (PM) and I will tell you anything I know.
Phil