I've been directed by LEOs to render armed assistance in the past, and I would say that it is entirely the discretion of the individual LEO.
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Anyways, I've received such direction from local Police Chiefs to Sheriffs to Military Police, and they pretty much all act the same. I am sure that it is predicated on how well they know the person they are ordering and whether they trust that person's judgment or not, and how imminent the situation at hand was. I can tell you in a firefight they are going to appreciate any qualified help they can get if it is going against them.