In most less severe cases of amnesia the people who have amnesia do know they have the amnesia and fully recover within a few weeks to a few months.
In more severe cases of amnesia the amnesia may be lifelong and the person may not know about the amnesia they have.
People with slight amnesia typically know they have the amnesia even when the loss of memory is through physical brain damage or psychological trauma.
If the brain damage that caused the amnesia is more severe and the person has near total or total brain damage then those people will unlikely know they have amnesia.