I think a rambling pink rose over a split wood fence is just gorgeous. That won't give you the privacy you are looking for though.

Ideally I'd think you want a variety of shrubs, evergreen and deciduous, rather than a row of one thing. You can plant perennials in between to give you a bit more privacy during the summer. A rambling rose would be nice. Red-twig dogwood (I like the one with the variegated leaf), yellow-twig dogwood, shrub roses, all kinds of vibernum, hydrangeas, azaleas (The species azaleas have a particularly nice open form), spireas, choysias. . . Underplant with hostas and interesting ground covers. With a little planning you can have something in bloom or displaying berries or colored foliage pretty much all year 'round.
If you want to go with native plants consider Oregon grape (the berries are edible), flowering currant (ditto), mock orange, Nootka rose, salal, high bush cranberry, native rhodies, spirea densiflora, evergreen huckleberries.

What kind of soil conditions, exposure, climate regime?

How wide would the bed along the fence be? Would a camellia work? I like my variegated leaf Red Twig Dogwood (like trolley molly), but it gets wide in a hurry. My "Winter Flame" dogwood grew quickly, too. Another favorite is Barberry, but the kind I have has thorns (don't know if all of them do). Of course, it all depends on the things bboy mentioned: soil conditions, exposure, climate regime.

To me, split rail fences cry out for climbing roses. They bloom for so long and so gloriously that you need a few in the row for sure. I would definately plant some taller shrubs and mix evergreens and deciduous shrubs and perennials in too.