Following the destruction of the Temple, R. Yoḥanan b. Zakkai ordained that the shofar be blown on Shabbat wherever there was a sitting beit din. He was referring to the main beit din of the time, where the new moon was sanctified, and whose judges had been ordained in an unbroken chain since the times of Moshe Rabbeinu. Wherever such a court was located, whether in Yavneh or elsewhere, the shofar was blown. If the court was no longer sitting there, it was prohibited to blow the shofar on Shabbat (MT, Laws of Shofar 2:9).
Rif, one of the greatest Rishonim, maintained that even after the chain of ordination was broken, the shofar should be blown in any important beit din. In fact, Rif followed this practice himself. In his beit din, the shofar was blown on Rosh Ha-shana even on Shabbat. However, the rest of the Rishonim disagreed with him, maintaining that the shofar was only to be blown in a beit din composed of those ordained in an unbroken chain from the time of Moshe. Therefore, there is no longer any beit din in whose presence the shofar can be blown. Furthermore, even the most devoted disciples of Rif did not follow his practice.
About a hundred years ago, after the rebuilding of Jerusalem had begun, R. Akiva Yosef Schlesinger wanted to blow the shofar in Jerusalem when Rosh Ha-shana was on Shabbat. His rationale was that the original law permitting shofar-blowing in Jerusalem and its environs was still in effect even after the destruction. Additionally, we saw that the Sages did not wish to put an absolute end to blowing the shofar when Rosh Ha-shana was on Shabbat. Thus, he felt that in his time, when there were no ordained judges, the shofar must be blown in the beit din of Jerusalem. Furthermore, since the mitzva of blowing the shofar is from the Torah, and the prohibition of blowing on Shabbat is rabbinic, whenever doubt arises we should blow the shofar, thus fulfilling the Torah commandment. R. Schlesinger also took great pains to explain how to implement shofar-blowing in a way that would eliminate the risk of people carrying on Shabbat. Although several rabbis expressed limited support for his position, those who disagreed prevailed, and the shofar was not blown on Shabbat. The main reason is as we established: License to blow the shofar on Rosh Ha-shana which is on Shabbat is limited to a beit din of ordained judges that is the central court of the time. The fact that none of the Torah greats who have lived in Jerusalem since the destruction have blown the shofar on Shabbat supports this understanding (Ir Ha-kodesh Ve-hamikdash 3:20).