The assessment date for ownership is January 1st, which is six months prior to the start of a new fiscal year (FY). In your case, the property was assessed to the owner of record on January 1, 2019. That assessment date applies for all of FY 2020 and the ownership may not be changed until January 1, 2020, which will affect bills issued for FY 2021.
As a courtesy, the Tax Collector has historically sent a copy of the bill to the new owner. Unfortunately, it takes about a month for the Assessors to be notified by the Norfolk County Registry of Deeds of a conveyance. If your deed was sent to the Assessors after the bills were issued, it is likely that you did not receive a copy of the bill currently due. Usually attorneys will remind you of the impending tax bill at the closing of the property, or your mortgage lender will escrow the amount needed for the next payment and take responsibility for making it.