If I see police in a shootout and I have a gun on me, am I legally allowed to join the fight against the criminal(s)?

Legally? Yes it is legal. You are legally authorized to use deadly force to protect yourself or a third party when threatened with imminent death or grievous bodily harm.

The police do not have any special ‘right’ to use deadly force, they simply have more legal protections for doing so. A private citizen involved in a shooting is very likely to automatically become a criminal suspect, where a law enforcement officer will face a review board before criminal charges against the officer are even considered. Police also face different legal standards in many jurisdictions (such as requiring ‘malice’ on the part of the officer.)

As a practical matter, ‘joining the fight’ is unlikely to be tactically sound or useful, and has a lot of potential to further complicate an already complicated situation. That is not to say that it hasn’t been done:

“…Approximately 20 minutes after first shooting from the observation deck, Whitman began to encounter return fire from both the police and armed civilians… Accompanied by 40-year-old civilian Allen Crum—whom the trio encountered as they ran toward the tower—they were the first to reach the tower's observation deck…” - Charles Whitman -

Source: https://www.quora.com/If-I-see-polic...the-criminal-s

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