What do you mean by performance standards? Police, especially those in the larger jurisdictions, are notoriously bad shots. In most, but not all, states that actually trust their ordinary citizens, after a background check, to carry a concealed handgun, there is a shooting skill requirement, and a requirement to know the law on when use of deadly force is justified. Texas adds some material on resolving conflicts conflicts without use of force, and even avoiding conflicts. Some states dispense with one or the other. Texas requires both. I learned that you can use deadly force to resist excessive force (more than needed to effect an arrest, and likely to cause serious bodily injury or death) by a police officer under Texas law, by studing the law, not from the class. So at least in Texas, the law applies to both police and citizens. In general the police are not allowed to use deadly force in any circumstances where a private citizen would not be justified in using it. They can carry handguns unconcealed, and that's about the only difference. (Since to become police officer they go through an equivalent background check, and get firearms training, that is just a little, but not as much as you might think, better than a CHL holder gets. OTOH, many need it, because they haven't been part of the "gun culture" before they became peace officers, which is really too bad, but is only common in the larger cities, and probably even there not typical)
So yes, just anybody can pickup (i.e. bear) their gun and fire away, when the circumstances justify it. Why should they not? They have a right to defend themselves don't they?
Police mistankenly shoot the wrong person at much higher rates than private citizens. Saying that is not knocking th police necessarily, since they come up on a situtation not really knowing who is who. The private citizen that is being mugges, raped or otherwise attacked, or is being robbed knows full well who the bad guy is. Oh yea, in Texas you can use deadly force to protect property, in some cases. It may not be a good idea, you have to ask yourself if your CD or super duper sound system is really worth the hassle, or would you rather just collect the insurance money and upgrade a little? But if your body is threatened, Texas law recognizes that it's better to be tried by 12 than carried by six, and so usually dispenses with the trial. Even in defending property cases, when taken to a Grand Jury, are almost always No-Billed, which means the grand jury finds no evidence of a crime by the defender.
OTOH, Vermont, recognizing the RKBA, does not restrict anybody from carrying anything anywhere at any time for any (lawful) reason. Open or concealed, they don't care. They don't have a big, or any, problem with private citzens getting a wild hair and blasting away for just any old reason. If people who do so, they go to jail, it's called reckless endangerment.
Police exist to take most of the burden of keeping the peace away from the ordinary citizen. The citzenry did not give up their right or powers when they created the police force.
Up until the last century or century and a half, depending on location, there were no police to speak of. A sherrif, and maybe a deputy or two. Or a town marshall and likewise a deputy or two. For anyting requiring more manpower, the sherriff would call on the posse commitatus, which is just the militia, which in turn is just the people, wearing a different hat, and reporting to the sherrif rather than their own commanders and ultimately the governor of the state, or territory as applicable.