The simple answer is, that's what the law says. The more complex answer looks both at legislative actions and court rulings.
The relevant case for US federal law is Leeke v. Timmerman, 454 U.S. 83. The State Solicitor of South Carolina declined to pursue criminal charges against certain prison guards. Petitioner's sued, the Supreme Court then held that
The decision to prosecute is solely within the prosecutor's
discretion. Thus, a private citizen has no judicially cognizable right
to prevent state officials from presenting information, through
intervention of the state solicitor, that will assist a magistrate in
determining whether to issue an arrest warrant.
reaffirming the holding of Linda R. S. v. Richard D., 410 U.S. 614 that
a private citizen lacks a judicially cognizable interest in the
prosecution or nonprosecution of another
So there is a rule, at least applicable to federal cases.
It may be of interest to look at the court's reasoning that resulted in that rule. The court says (emphasis added):
The threshold inquiry is whether respondents have standing to
challenge the actions of petitioners. As in Linda R. S., there is a
questionable nexus between respondents' injury -- the alleged beatings
-- and the actions of the state officials in which they gave information to a Magistrate prior to issuance of an arrest warrant.
Even without the prosecutor's acts, there is no guarantee that
issuance of the arrest warrant would remedy claimed past misconduct of
guards or prevent future misconduct. Even if a prosecution could
remedy respondents' injury, the issuance of an arrest warrant in this
case is simply a prelude to actual prosecution. Respondents concede
that the decision to prosecute is solely within the discretion of the
prosecutor. It is equally clear that issuance of the arrest warrant
in this case would not necessarily lead to a subsequent prosecution.
A private citizen therefore has no judicially cognizable right to prevent state officials from presenting information, through
intervention of the state solicitor, that will assist the magistrate
in determining whether to issue the arrest warrant. Just as
respondents were able to present arguments as to why an arrest warrant
should issue, a state solicitor must be able to present arguments as
to why an arrest warrant should not issue. This is not a case in which
prison officials interfered with the transmittal of information from
respondents to the magistrate, thereby interfering with respondents'
ability under South Carolina law to seek the arrest of another.
On the other hand, in Washington state, private prosecution is allowed under CrRLJ 2.1(c), as affirmed recently by the state's Supreme Court: "Under the citizen complaint rule, '[a]ny person' may initiate
criminal proceedings". This article discussed some of the arguments and rulings related to the private prosecution rule (arguing against private prosecution), and this article compares the US vs. the UK where private prosecution is more viable. There does not seem to be a compelling constitutional reason to prohibit private prosecution, but each jurisdiction has the power to set its own procedural rules, therefore there can be variation.