The taxing unit does not prorate taxes for the year. Most contracts for the sale of real property provide that the taxes for the year of the sale will be prorated between the buyer and seller. The sales contract and any closing statement received when the property was sold should indicate how the tax liability was handled.
If the seller owned the property on January 1 of last year then the seller is liable for the taxes for all of last year. The taxing unit can proceed against the seller personally for the tax.
The taxing unit can also proceed against the buyer to foreclose the tax lien. The buyer should be informed of the taxes since the buyer has a vested interest in not having the tax lien foreclosed on the property.