The two former Tory leadership candidates are the first appointments Ms Badenoch has made to shadow the so-called “great offices of state”: the Foreign Secretary, Chancellor and Home Secretary.
Dame Priti and Mr Stride were both early rivals of Ms Badenoch during the Tory leadership contest.
The PA news agency understands the pair’s appointment to the opposition frontbench is an effort to unite different factions of the Conservative Party following the long internal election battle.
The new Conservative leader insisted she would offer all her rivals in the race roles in her shadow cabinet, but former home secretary James Cleverly has already ruled himself out from such a job.
Ms Badenoch had earlier named Laura Trott as her shadow education secretary and Tory former minister Neil O’Brien as a shadow education minister, ahead of education questions in the House of Commons.
It was reported on Sunday that Essex MP Dame Rebecca Harris would become the party’s new chief whip, while Nigel Huddleston and Lord Dominic Johnson have been made joint chairmen.