After October 1 of each year, cities and towns may levy taxes upon property and all subjects of taxation liable therefor at a rate not in excess of the constitutional limit upon assessments to be made by the city or town clerk or other person designated by the council or other governing body, such assessment to be made on the state assessment in the manner provided by the Constitution of the state or in the manner hereinafter authorized by law; provided, however, that any municipality may by ordinance provide that the tax year for such municipality shall commence on October 1 of each year and end on the next succeeding September 30, in which case cities and towns shall levy taxes as above set forth prior to August 1 of each year.

After the assessment has been made, it shall be returned to the council or other governing body which shall thereupon give 10 days’ notice by publication in a newspaper published in the city or town or, if no newspaper is published in such city or town, then by posting notices in three or more places in such city or town that the assessment has been completed and that the council or other governing body will hear and determine objections thereto upon a day not more than 30 days from the date on which said notice was directed to be made.

The council or other governing body may, however, authorize such assessment to be made by a board of assessors who, when the assessment has been completed, shall give a similar notice that such board will hear and determine objections to the assessment at a time and place designated in such notice not more than 30 days thereafter. On the day set for the hearing of objections, the council, or other governing body or board, as the case may be, shall hear such objections and determine the assessment.