As explained in Rashi, Rav Hirsch, and the Art Scroll Chumash, the words of the blessings and the curses were spoken by the Kohanim and Leviim in the middle.
The command on what to do and what to say was given in the Torah. When it was actually carried out, in Yehoshua 8:33](https://www.chabad.org/library/bible_cdo/aid/15792#showrashi=true&v=33) the navi emphasizes that they carried it out exactly as commanded.
33 And all Israel, and their elders and officers and their judges, stood on this side of the Ark and on that side, before the priests the Levites, the bearers of the Ark of the covenant of the Lord, the stranger as well as the native born, half of them over against Mount Gerizim and half of them over against Mount Ebal, as Moses the servant of the Lord had commanded, to bless the people of Israel first.
After this was done, Yehoshua read the Torah to the Bnai Yisrael so that they would hear everything that they had been commanded.
34 And afterward he read all the words of the law, the blessing and the curse, according to all that is written in the book of the Torah.
35 There was not a word of all that Moses commanded, which Joshua did not read before all the congregation of Israel, with the women, and the little ones, and the strangers that walked among them.
Part of what he read included the commandment about the blessing and the curse which had just been fulfilled. Since the actual words that had been said were in the Torah, they were included in the reading.