Police on a call can do many things that civilians can not do.
Not legitimately, no. As soon as you get away from the core value that "the police are the public and that the public are the police, the police being only members of the public who are paid to give full-time attention to duties which are incumbent on every citizen in the interests of community welfare and existence," you have an illegitimate use of police power. And one which violates the 14th Amendment's guarantee of equal protection.
If I witness a serious crime, I have the right (and perhaps the moral duty) to use reasonable and necessary force to stop it, same as a cop. (This is not legal advice, consult your local laws, I am discussing here what a sane and just set of laws should look like.) I don't have the right to break into someone's house to look for evidence, or to arrest someone accused of a crime that happened somewhere else days ago -- but neither does a cop, that power comes from a specific warrant, not a badge.
Yes, I don't have the right to enforce traffic laws on the public roads, but the roads are government property and they get to hire whatever security they want for that. Traffic cops are to the roads as "mall cops" are to malls -- empowered by the management, nothing more.
Remember that this country is older than the idea of police as we know them today. We got by all right for quite a while without special rights for police.