Traditional standing requirements in federal courts, are doctrinally derived from Article III of the U.S. Constitution which governs the federal judicial power.
State constitutions or state common law have been invoked to apply substantially identical standing requirements in state court in most kinds of cases, but sometimes with narrow exceptions for advisory opinions in connection with the legislative process. Under Texas precedent, Spence v. Fletchler, and subsequent Texas cases, even a "bounty" type scheme that allows parties without standing to bring certain kinds of lawsuit, is constitutionally permitted under Texas law.
The U.S. Supreme Court in the case of Whole Women's Health v. Jackson, applies a standing test to the federal court challenge of the state law, but the majority opinion is agnostic as to state requirements to show standing or something equivalent that Texas S.B. 8 does not itself require. As some pertinent parts of the Official Syllabus to the opinion explain:
(3) The petitioners name other defendants (Stephen Carlton, Katherine
Thomas, Allison Benz, and Cecile Young), each of whom is an executive
licensing official who may or must take enforcement actions against
the petitioners if the petitioners violate the terms of Texas’s Health
and Safety Code, including S. B. 8. Eight Members of the Court hold
that sovereign immunity does not bar a pre-enforcement challenge to S.
B. 8 against these defendants. Pp. 11–14.
(4) The sole private defendant, Mr. Dickson, should be dismissed.
Given that the petitioners do not contest Mr. Dickson’s sworn
declarations stating that he has no intention to file an S. B. 8 suit
against them, the petitioners cannot establish “personal injury fairly
traceable to [Mr. Dickson’s] allegedly unlawful conduct.” See
California, 593 U. S., at ___ (slip op, at 9). P. 14.
(c) The Court holds that the petitioners may bring a pre-enforcement
challenge in federal court as one means to test S. B. 8’s compliance
with the Federal Constitution. Other pre-enforcement challenges are
possible too; one such case is ongoing in state court in which the
plaintiffs have raised both federal and state constitutional claims
against S. B. 8. Any individual sued under S. B. 8 may raise state and
federal constitutional arguments in his or her defense without
Whatever a state statute may or may not say about a defense,
applicable federal constitutional defenses always stand available when
properly asserted. See U. S. Const., Art. VI. Many federal
constitutional rights are as a practical matter asserted typically as
defenses to state-law claims, not in federal pre-enforcement cases
like this one. See, e.g., Snyder v. Phelps, 562 U. S. 443 (First
Amendment used as a defense to a state tort suit). Other viable
avenues to contest the law’s compliance with the Federal Constitution
also may be possible and the Court does not prejudge the possibility.
Justice Sotomayor's dissenting opinion in Whole Women's Health argues that if the State delegates unsupervised authority for private individuals to enforce state law in matters in which they have no standing, that it may do so but that this has consequences. She reasons that when a state does this, that the state has effectively hired these private individuals to be state government officials who are bound by injunctions against any state official even if they are not personally involved in the injunction litigation (Slip. Op. at 7-9):
No party has identified any prior circumstance in which a State has
delegated an enforcement function to the populace, disclaimed official
enforcement authority, and skewed state-court procedures to chill the
exercise of constitutional rights. Because S. B. 8’s architects
designed this scheme to evade Young as historically applied, it is
especially perverse for the Court to shield it from scrutiny based on
its novelty.<3> . . . <4>.
<3>The Court responds by seizing on my mention of S. B. 8’s chilling
effect. Ante, at 16. No one contends, however, that pre-enforcement
review should be available whenever a state law chills the exercise of
a constitutional right. Rather, as this Court explained in Young,
pre-enforcement review is necessary “when the penalties for
disobedience are . . . so enormous” as to have the same effect “as if
the law in terms prohibited the [litigant] from seeking judicial
construction of laws which deeply affect its rights.” 209 U. S., at
147. All the more so here, where the State achieves its unconstitutional aim using novel procedural machinations that the
Court fails to acknowledge.
<4> The Court also holds that the Texas attorney general is not a
proper defendant. For the reasons explained by THE CHIEF JUSTICE,
ante, at 2– 3, this conclusion fails even under the Court’s own logic.
The Court further observes that “no court may ‘lawfully enjoin the
world at large.’” Ante, at 10–11 (quoting Alemite Mfg. Corp. v. Staff,
42 F. 2d 832 (CA2 1930)). But the petitioners do not seek such relief.
It is Texas that has taken the unprecedented step of delegating its enforcement authority to the world at large without requiring any
pre-existing stake. Under the Court’s precedents, private actors who
take up a State’s mantle “exercise . . . a right or privilege having
its source in state authority” and may “be described in all fairness
as . . . state actor[s].” Edmonson v. Leesville Concrete Co., 500
U. S. 614, 620 (1991). This Court has not held that state actors who
have actual notice of an injunction may flout its terms, even if it
nominally binds other state officials, and it errs by implying as much
now. The Court responds by downplaying how exceptional Texas’ scheme
is, but it identifies no true analogs in precedent. See ante, at 11
(identifying only “somewhat” analogous statutes). S. B. 8 is no tort
or private attorneys general statute: It deputizes anyone to sue
without establishing any pre-existing personal stake (i.e., standing)
and then skews procedural rules to favor these plaintiffs.
This analysis is a twist on the analysis discussed by @Fizz in an answer looking a the standing issue in some qui tam actions as a question of how much authority a legislative body may delegate (in that case, in a federal litigation context).