Its role is as a strengthener used to emphasize the fact that the speaker and the listener are one and the same. I suppose it's passable to omit the strengthener for the 1st and for the 2nd person, but you need to use the strengthener for the 3rd person. Here are a few examples:
- "Eu vorbesc cu tine." = "I talk to you."
- "Eu vorbesc cu mine (însumi)." = "I talk to myself."
- "Tu vorbești cu mine." = "You talk to me."
- "Tu vorbești cu tine (însuți)." = "You talk to yourself."
- "El vorbește cu el." = "He talks to him." (him being some other guy!)
- "El vorbește cu el însuși." = "He talks to himself."
I didn't cover feminine/plural combinations, but hopefully this was enough to illustrate the point.