This is the method for iD I believe as gleaned from a previous similar question… http://forum.openstreetmap.org/viewtopic.php?pid=369655
Click on the road to select it.
Double click on an empty place on the selected road where the split is required and a node, small circle, is added. (unless you already have a node in the correct spot).
Click on the node to select it, a “palette” appears with one of the options a pair of scissors. Clicking on the scissors splits the road segment into two segments.
You should then be able to do the same with the other end of the private road segment and add the tag access=private to the middle bit if still there or delete it if the road is not there. Also add a gate or fence at the ends if present.