The school board officials’ petition seeking Supreme Court review in the case, O’Connor-Ratcliff v. Garnier, No. 22-324, said their accounts were personal and were created and maintained “without any direction, funding, support or other involvement by the district.”

The petition added, “A public official does not engage in state action when blocking users from a social media account where, as here, the account is not operated pursuant to any governmental authority or duty.”

The second case, Lindke v. Freed, No. 22-611, concerned a Facebook account maintained by James R. Freed, the city manager of Port Huron, Mich. He used it to comment on a variety of subjects, some personal and some official. Among the latter were descriptions of the city’s responses to the coronavirus pandemic.

The posts prompted critical responses from a resident, Kevin Lindke, whom Mr. Freed eventually blocked. Mr. Lindke sued and lost. Judge Amul R. Thapar, writing for a unanimous three-judge panel of U.S. Court of Appeals for the Sixth Circuit, in Cincinnati, said Mr. Freed’s Facebook account was personal, meaning that the First Amendment had no role to play.

“Freed did not operate his page to fulfill any actual or apparent duty of his office,” Judge Thapar wrote.” And he did not use his governmental authority to maintain it. Thus, he was acting in his personal capacity — and there was no state action.”

Katie Fallow, a lawyer with the Knight First Amendment Institute at Columbia University, which had sued Mr. Trump, said in a statement that the question the justices agreed to decide had only become more pressing.

“With more and more public officials using social media to communicate with their constituents about official business,” she said, “public officials’ social media accounts are playing the role that have historically been played by city council meetings, school board meetings and other offline public forums.”