Whenever any lands belonging to the county are included within a weed district, the county legislative authority shall determine the amount of the assessment for which the lands would be liable if they were in private ownership, and the county legislative authority shall appropriate from the current expense fund of the county sufficient money to pay such amounts. Whenever any state lands are within any weed district, the county treasurer shall certify annually and forward to the appropriate state agency for payment a statement showing the amount of the assessment to which the lands would be liable if they were in private ownership, separately describing each lot or parcel and, if delinquent, with interest and penalties consistent with RCW 84.56.020.