That’s what is brought in the link you sent me on the subject:
https://daat.ac.il/mishpat-ivri/skirot/311-2.htm
And the question is: is this authority held by every religious court? According to Tosafot, from the verse “And these are the ordinances that you shall place before them” (Exodus 21:1), it is to be derived, based on the Sages’ exposition, that not only does adjudication require expert judges, but also “for any act of enforcement and coercion there is a need for expert religious court judges [= ordained judges].” However, the author of Ketzot HaChoshen already wrote that nowadays, when we have no experts, the religious court acts by virtue of the agency of the earlier ones, just as it acts by their authority in other matters as well14.

According to this, not every individual has permission to compel another to fulfill commandments, but only a religious court15.

14. See Ketzot HaChoshen, sec. 3, subsec. 1.
15. But see the gloss of Netivot HaMishpat there, subsec. 1: “Since this is similar to a case of making a sukkah and he does not do so, where we compel him to fulfill the commandment, every person is commanded to separate his fellow from prohibition, even one who is not part of a religious court, as is proven in Bava Kamma 28a regarding a pierced slave whose term has ended [and he is forbidden to a Canaanite maidservant, yet refuses to go free], that his master may strike him in order to separate him from the prohibition of the maidservant.” And the author of Ketzot HaChoshen replied to his words, in his book Meshovev Netivot, there, subsec. 1, that one should distinguish between separating someone from a prohibition and coercing fulfillment of positive commandments. Regarding every person’s duty to separate others from prohibitions, see my article, “Religious Legislation in a Secular Society,” Machanayim 13 (1995), p. 278.

Now I see that there is a distinction between coercion regarding a prohibition and coercion regarding a positive commandment (where a religious court is needed). And maybe regarding the question I asked about coercion in “do not stand idly by your neighbor’s blood,” one has to say that because it involves positive action, it is considered like a positive commandment and requires a religious court. But it’s still difficult regarding what you said about destroying the book, because a person can possess a book that is forbidden for benefit without benefiting from it, and then there is still no permission for someone else to destroy the book. As for idolatry, the very possession of it may itself be a prohibition (and not only its use).

And maybe one also should have mentioned there the distinction between coercing a secular person and coercing a religious person, since with a secular person coercion is not applicable (and then there’s no need to reach the principle of “it doesn’t sit right”).

I think the intention there is to acts of enforcement and coercion of a legal ruling. That’s why this is a case of acting by their agency.
In any case, in the accepted understanding, every person may compel another person regarding commandments. Even if the term “religious court” is mentioned, it’s like “if a minor is eating forbidden carcasses, the religious court is commanded to separate him,” where the intention is any person who sees this, acting as an agent of the religious court (or of the public. It is really an obligation on the public).

But in Netivot HaMishpat it says explicitly:
“Since this is similar to a case of making a sukkah and he does not do so, where we compel him to fulfill the commandment, every person is commanded to separate his fellow from prohibition, even one who is not part of a religious court.”

If there is no difference between every person and a religious court, then what is he coming to add here?

What is written there is exactly what I wrote to you here: that the difference between ordained judges and ordinary people is only in a religious court’s coercion to enforce its rulings, but regarding coercion about commandments, every person coerces just like a religious court. That itself is what he is innovating there. Admittedly, my proof from the master of the slave can of course be rejected, since after all he is his master, but this is not the place to elaborate.

And in Meshovev there he distinguished between a positive commandment and a prohibition, and also within a prohibition between an ordinary person and a religious court (which can strike him until his soul departs).