Assuming that witness testifies for defence.
A background check could mean many different things. It's not impossible someone investigating your background could learn of it. But that doesn't mean it's likely. If you were the defendant, rather than a witness, it would be likely to be discovered.
Imagine that the defendant is convicted, and appeals the conviction. The appellate court writes an opinion and refers to the testimony heard at trial. The witness's name might be mentioned in the opinion and the opinion might be published as a "published opinion." That would mean someone doing a google search could well find the opinion.
A device called a "subpoena" protects witnesses because it compels them to come to court and give testimony. When asked why they gave testimony, they may truthfully answer, "I had no choice. I received a subpoena." Witnesses are generally subpoenaed to give testimony.
Not legal advice as I don't practice law in Washington. It's just my two cents on the facts you describe in light of general principles of law. If you need legal advice, please consult a lawyer who holds Washington licensure.
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