In my state, the purchaser is responsible for the "supplemental tax," where the property tax assessment is "trued up" on the purchase price. The escrower only pays the regular, secured property taxes out of escrow.

Occasionally buyers here are surprised by this. I had to pay 6 months of stepped up tax basis on my own after buying, but I knew it was coming and had budgeted it. Then you know how much your property tax will go up every year, thank you Prop 13. . .