In selecting Mr. Priebus, Mr. Trump passed over Stephen K. Bannon, a right-wing media provocateur. But the president-elect named Mr. Bannon his senior counselor and chief West Wing strategist, signaling an embrace of the fringe ideology long advanced by Mr. Bannon and of a continuing disdain for the Republican establishment.
The dual appointments — with Mr. Bannon given top billing in the official announcement — instantly created rival centers of power in the Trump White House.
Mr. Bannon’s selection demonstrated the power of grass-roots activists who backed Mr. Trump’s candidacy. Some of them have long traded in the conspiracy theories and sometimes racist messages of Breitbart News, the website that Mr. Bannon ran for much of the past decade.
The site has accused President Obama of “importing more hating Muslims”; compared Planned Parenthood’s work to the Holocaust; called the conservative commentator Bill Kristol a “renegade Jew”; and advised female victims of online harassment to “just log off” and stop “screwing up the internet for men,” illustrating that point with a picture of a crying child.
The grass-roots activists may be angered by the selection of Mr. Priebus as chief of staff, viewing him as a deal maker who will be too eager to push the new president toward compromise on issues like taxes, immigration, trade, health care and the environment.
In a statement Sunday afternoon, the transition team emphasized that the two men would work “as equal partners to transform the federal government.”
The arrangement appeared aimed at ensuring that both men would be required to sign off on many decisions jointly. And Mr. Bannon was assured that he reports directly to Mr. Trump, not to Mr. Priebus.
The simultaneous announcement and competing lines of authority are consistent with Mr. Trump’s management style in his businesses and in his campaign: creating rival power structures beneath him and encouraging them to battle it out.
It is also a reflection of who has the ear of the president-elect: his children, and especially his eldest daughter, Ivanka Trump, and her husband, Jared Kushner. Both had argued that the chief of staff job should not be held by someone too controversial, according to several people familiar with the decision-making inside the transition effort.
Mr. Kushner is likely to wield great influence over the new president regardless of whether he holds a formal title. Mr. Kushner, who has no experience in politics or government, often has the final word in advising Mr. Trump.
But while Mr. Trump apparently feels comfortable with Mr. Priebus, the people with knowledge of his weekend decision said Mr. Bannon was still the adviser who was better able to talk forcefully to the president-elect during difficult moments.
The transition team appeared eager to appease concerns among Mr. Trump’s most fervent supporters that choosing Mr. Priebus meant the president-elect had already caved to the Washington “swamp” he had promised to drain. The team also wanted to mollify Mr. Bannon, and to that end, the official statement mentioned Mr. Bannon first.
“We had a very successful partnership on the campaign, one that led to victory,” Mr. Bannon said in the statement. “We will have that same partnership in working to help President-elect Trump achieve his agenda.”
Mr. Priebus said he looked forward to working with Mr. Bannon and Mr. Trump “to create an economy that works for everyone, secure our borders, repeal and replace Obamacare and destroy radical Islamic terrorism.”
Mr. Priebus is expected to have multiple deputies, including Katie Walsh, the chief of staff of the Republican National Committee, who is close to Mr. Priebus and helped ensure a tight working relationship between the party’s operational infrastructure and Mr. Trump’s campaign.