Solutions Institute states: "We could not find an official Washington citizen’s arrest statute. We called their legislative information center, and their law library, and both informed us that the statute does not exist. A 2005 report from the Department of Licensing confirms this..."
Department of Licensing: "Summary of Citizen Arrest" (2005) reproduced here:
"Citizen Arrest in Washington State
Under Washington law, a private person can conduct a citizen’s arrest for a misdemeanor if the misdemeanor: (1) was committed in the citizen’s presence and (2) constituted a breach of the peace. State v. Gonzales, 24 Wn. App. 437, 439, 604 P.2d 168 (1979); Guijosa v. Wal-Mart Stores, 101 Wn. App. 777, 791, 6 P.3d 583 (2000).

A person can also conduct a citizen’s arrest for felonies. See State v. Malone, 106 Wn.2d 607, 724 P.2d 364 at FN1 (1986) citing State v. Miller, 103 Wn.2d 792, 698 P.2d 554 (1985) and State v. Gonzales , 24 Wn. App. 437, 604 P.2d 168 (1979).

RCW 10.04.020 provides for a citizen’s arrest at the direction of a district court judge, as follows: Arrest -- Offense committed in view of district judge.

When any offense is committed in view of any district judge, the judge may, by verbal direction to any deputy, or if no deputy is present, to any citizen, cause such deputy or citizen to arrest such offender, and keep such offender in custody for the space of one hour, unless such offender shall sooner be taken from such custody by virtue of a warrant issued on complaint on oath. But such person so arrested, shall not be confined in jail, nor put upon any trial, until arrested by virtue of such warrant.

While Washington has no statute concerning citizen’s arrests generally, RCW 9A.04.060 provides that the common law is applicable where not repugnant to the provisions of the state constitution or statutes. Gonzales, 24 Wn. App. at 439; State v. Miller, 103 Wn.2d 792, 795, 698 P.2d 554 (1985).

Application of the common law doctrine of citizen’s arrest is demonstrated in the synopsis of the case below:

Instruction on lawful arrest in prosecution for third degree assault, arising from defendant's resistance to apprehension by store personnel, did not improperly state the law; although language was taken from statute which sets forth defense available to store personnel who have detained a suspected shoplifter, statutory language was consistent with common-law right of citizen arrest
which permits detention of a suspected shoplifter on reasonable grounds. State v. Jones (1992) 63 Wash.App. 703, 821 P.2d 543, review denied 118 Wash.2d 1028, 828 P.2d 563. Assault And Battery 96(3); Criminal Law 808.5-"

Similar to other states, there is indeed statute law expressly permitting private persons to perform an arrest if for extradition purposes:

2017 Revised Code of Washington Title 10 - CRIMINAL PROCEDURE 10.88 Uniform criminal extradition act. 10.88.330 Arrest without warrant.

Universal Citation: WA Rev Code § 10.88.330 (2017)

RCW 10.88.330 Arrest without warrant.
(1) The arrest of a person may be lawfully made also by any peace officer or a private person, without a warrant upon reasonable information that the accused stands charged in the courts of a state with a crime punishable by death or imprisonment for a term exceeding one year, but when so arrested the accused must be taken before a judge or magistrate with all practicable speed and complaint must be made against him or her under oath setting forth the ground for the arrest as in RCW 10.88.320; and thereafter his or her answer shall be heard as if he or she had been arrested on a warrant.complaint must be made against him or her under oath setting forth the ground for the arrest as in RCW 10.88.320; and thereafter his or her answer shall be heard as if he or she had been arrested on a warrant.