Indeed, legally, private individuals and companies are allowed to act in emergencies. For example, I generally should not break into my neighbor’s home. However, I am legally allowed (and morally obligated) to forcibly enter their residence if their house is on fire, or they’re being attacked by a burglar, etc. and I am able to prevent some of the harm.
Of course, if we assume we’re talking about situations where the government needs a warrant, the legality becomes more complicated. At what point does something become an emergency? I would say it’s not an emergency if there is time to inform the government and to let the government prevent the injury. If we assume the government is unwilling or unable to act, then the window for action should expand by some measure.
Private companies having arrest rights is just a nonstarter of an idea (putting it kindly).
https://en.wikipedia.org/wiki/Citizen%27s_arrest#United_Stat...
However, they look very different. The major distinction is that when a private party requests an injunction allowing them to e.g. trespass on their neighbor’s land, the court will require notice and a hearing for the defendant. So, if a chemical plant needs to do earth works on a neighbor’s land to prevent a collapse, etc. the judiciary may well issue an order requiring the neighbor to let the company enter.
Frankly, notice and hearing should probably be required for some criminal warrants too. I can think of a few indictments and arrest warrants that have recently been issued where there is a genuine question as to probable cause and the alleged illegality of the conduct. It’s not fair for people who are not a flight risk to be arrested (and often imprisoned) with no opportunity to defend themselves.