Upon receiving such petition, the selectboard shall give 10 days' notice of a hearing upon the public necessity for the construction of such sewer by posting notices to that effect in three public places in such village. When, after notice and hearing, the selectboard decide that the public good requires the construction of such sewer, they may proceed to construct the same at the expense of the town. They then shall notify all parties who will be benefited thereby to appear before them to be heard upon the apportionment of the cost of such sewer among those benefited, by giving to each of such persons a written notice of the time and place of such hearing, signed by a majority of the selectboard, which notice shall be given not less than 10 days before such hearing. All parties benefited by the construction of such sewer shall be assessed for the construction of the same according to the benefit received.