Can you add the same hit as someone else?
You were correct

There's two key phrases in the Hit rule to support this,


  You make a hit by putting a card directly on a Phase already laid down. The card must properly fit with the cards already down.


The first phrase deals with your cards being added to the played Phase and not a separate "also could legally go here" pile. In effect, your card becomes part of the phase. This leads to the second part: since your card is now part of the Phase, a duplicate would not properly fit (at least in this example of a straight Phase). 

Your opponent can still play the 10 because that will properly fit the run, and either of you could subsequently add a 3, or 11, etc.