The signature is the return's declaration. Penalty of perjury means the signer states, under that penalty, examination of the filing and belief in its truth: true, correct, and complete to the best of their knowledge and belief. That wording sets the standard the signer is held to — examined belief, reaching every attachment — and it is why a return needs real signatures rather than informal assent.
Joint returns extend the rule to both spouses: each signs a joint amended return for it to count as valid. Paid preparation adds a second signature in its own area: anyone compensated to prepare, assist, or review must sign and enter a Preparer Tax Identification Number (PTIN). An unpaid helper signs nothing. The two signatures attest different things — the taxpayers' to the return's truth, the preparer's to the preparation — so one can never stand in for the other.
Worked example
June pays a preparer to complete her joint amended return with her husband Ray. Three signatures belong on the filing: June's and Ray's as the joint filers, plus the preparer's in the paid-preparer area with PTIN. If Ray is unreachable, June cannot simply sign twice and file; the joint return needs his signature (through proper authorization if he cannot sign), and the preparer's signature covers preparation rather than substituting for Ray.