For those interested in the gory details, please see the full decision in Trump v. Vance. The first half of the linked document contains the Court's majority opinion and the concurring opinion of Justices Kavanaugh and Gorsuch, followed by the separate dissenting opinions of Justices Thomas and Alito.
Based on a quick read, it's apparent that Justice Thomas agrees with the general court's opinion that a sitting President "is not entitled to absolute immunity from issuance of [a] subpoena". Thomas dissents on the grounds that the President "may be entitled to relief against [a subpoena's] enforcement" based on his special status as president. Thomas argues (agreeing with the president) that the case should be vacated and returned to the lower court, so that the President can argue that his duties to the nation take up all of his time and attention and he thus deserves relief from the subpoena.
Justice Alito makes a stronger dissent. While he agrees that a sitting President is not absolutely immune from a state subpoena, he argues that such a subpoena should not be enforced unless it meets a stringent legal test, such that:
- It does not interfere with the President's execution of his duties of office, and
- It carries a weight of urgency — e.g., subpoenaing information related to suspected treason — as a guarantee it is not merely issued to harass.
In the absence of those conditions, in Alito's opinion, the President is only subject to the constitutional process of impeachment and trial in Congress.
So no, neither dissenting Justice suggests that the President is absolutely immune from this kind of subpoena. Instead, they are concerned that the issuance of such subpoenas might negatively impact on the President's ability to perform the duties of the office, and argue that the President might legitimately be granted varying degrees of relief from such legal actions at a court's discretion.
