I do both. I'm right-handed, so I hold the knife in my right hand and brush the left side (the side near me) with the edge pointing at me, and the right side (the side away from me) with the edge pointing away.
For the near side, I move the pad of my finger along the spine of the blade, moving from the handle to the tip. This means the edge of the blade moves backwards up my finger and can't cut me. I also arch my finger a bit so the edge isn't really touching my finger.
For the far side, I keep the pad just above the edge of the blade, pushing slightly out as I run my finger towards the tip.
In either case, if my finger moves and I miss some food, I take my finger off -- moving it down the blade towards the edge -- then start over, rather than trying to reset while half a millimeter from the edge.
I do occasionally get tiny microcuts in the skin this way, but I've never had a major cut from it. I find it easier than switching hands to clear each side, and safer than trying to clear the near side with my left-hand fingers pointing down the blade (which requires odd contortions or twisting the knife so my finger is under it and I can't see what I'm doing).
On the other hand, I picked up a stack of salad bowls two weeks ago and had blood spurting everywhere because someone set the bowls down too hard and broke one, and the sharp edge of the broken bowl put two giant slashes in my finger. Safety is relative, I guess.