Say your out and about, and you happen to see an assult underway, where it appears to you the assilant is threatening someone's life. Are you allowed to pull your weapon, order the assaliant to cease and hit the ground, and if he continues his attack you can shoot him?
Now your not playing Cop, just using force to save someone elses life, rather than your own.
In most states, if a third party would be legally justified in using lethal force to defend himself, then you would be legally justified in using lethal force to defend that person. The law varies from state to state, so you would need to speak to a lawyer in your state.
I would disagree with you when you say "Now you're not playing Cop..." That is precisely what you would be doing in such a situation and it is a very dangerous thing to do.
Intervening in a third party situation is very risky. First, there is the immediate physical risk. A gun is not a talisman. It does not guarantee that you will prevail. You may lose. The perp may be able to close the distance and disarm you. The perp may have a gun and get a lucky shot off. You may miss. You might not have seen the perp's partner behind you. Are you willing to risk your life for someone you don't know?
Second, there is the legal risk. It often isn't clear just what is really going on in such situations.
For example, suppose you are at the local strip mall when you see a man burst out of store, running in terror. Seconds later, another man runs out with a gun in his hand, chasing the first man. What have you just seen? A nutcase trying to kill a shopper? The store owner chasing a robber? An undercover officer chasing a suspect?
If you use deadly force in a situation where the person who you thought was the victim was, in fact, not legally justified in using force to defend themselves, then you could face murder charges. Are you willing to risk going to jail for the rest of your life for someone you don't know?
I carry on a regular basis. I won't say that I would never intervene in a third-party situation. But my default choice would be to get out of Dodge and call the police.
My gun is to protect myself and my family. I don't get paid to save other people. I don't have backup on the way. I don't have a radio to the dispatcher. I'm not wearing a blue uniform that will identify me to the responding officers as one of the good guys. I don't have soft body armor. If I end up dead, the victim isn't going to provide for my family. If I end up in court, the victim isn't going to pay my legal expenses (which could top $100,000 in the case of a murder charge).