The company issuing the bonds pays interest for the period they had use of the funds. Amortization is recorded with interest expense when interest is paid or accrued.  If the bond issue date is the same as the interest payment date they would not have any accrued interest on that date.

However, if you are referring to bonds that are sold between interest payment dates, the investor pays the accrued interest when they purchase the bonds. On the interest date they receive payment for the entire period. The issuer accounts for the purchase by deducting the accrued interest from the bond proceeds and recording a discount/premium for the difference.