A criminal history check may reveal that you were arrested and charged and that the case was dismissed. You can run a criminal history on yourself at the Washington State Patrol website to see what your criminal history looks like. A motion can be made to expunge the case from your history if you have been crime-free and two years have elapsed since the dismssal date (or three years from the arrest date).
A compromise of a misdemeanor will not show up on most background checks because it results in a dismissal of the case without conviction. This is one of the reasons it is a great way to resolve a theft case.
As stated above, you can petition WSP to remove the arrest information from it's database. It is advisable to do this as soon as possible as any public record can be found with a thorough enough search. In addition to the requirement that you wait for two years from the date the court adopted the misdemeanor compromise (or three years from the date of the arrest), WSP has the option to deny your request if (1) the disposition was a completed deferred prosecution. (2) you have prior convictions for gross misdemeanor or felony, or (3) you have been arrested for, or charged with, a subsequent crime. If this is your only run-in with the law it usually is not an issue getting the info removed.
The link for petition for deletion of arrest record is included below.