What Type of Annuity is Involved? That Dictates the Answer
What is really important is the type of annuity the person purchased.
Life annuities pay money each month but only for so long as the named annuitant lives. On his or her death, the payments cease. With that type of annuity no beneficiary is named as it ends on the person's death.
Some annuities provide for payments during the lifetime of the annuitant and also guarantee that if the annuitant dies before a set period (such as 10 years) payments will continue for the remainder of those 10 years to a named beneficiary. If that's they type you would need to name a beneficiary who would receive the funds for the remainder of the period.
Other annuities are "deferred" annuities where the funds merely accumulate and payments are deferred until the owner elects for them to start. With such annuities a beneficiary is essential.