One of the Arc Hydro processing steps creates Drainage Points. This gives you two points near each junction. They are not on top of each other.
The other thought I had was to use Data Management Tools.tbx\Features\Feature Vertices To Points to created points at the ends of each line feature. The assumption here is that the upstream end of your stream polygon is snapped to the downstream end of the next polygon.
To eliminated duplicates (points on top of each other) use Data Management Tools.tbx\General\Delete Identical w/o specifying any fields to compare.