Here the word "us" refers to Peter, the other apostles, and all of the Jewish believers who had received the Holy Spirit at Pentecost. (See: [[rc://en/ta/man/translate/figs-inclusive]])
Peter leaves out some things to keep the story short. Alternate translation: "the Holy Spirit came on the Gentile believers, just as he came on the Jewish believers at Pentecost" (See: [[rc://en/ta/man/translate/figs-ellipsis]])