Following the laws of prayer, the Shulhan Aruch deals with the laws of synagogue. The Shulhan Aruch (Orah Haim 150:1) rules that residents of a town have an obligation to build a synagogue, and the Beur Halacha (ibid.) extends this obligation to establishing a Bet Midrash as well. As well, the Zohar (Beshalah) says that since a synagogue is considered a microcosmic Bet HaMikdash, one fulfils the obligation to rebuild the Bet HaMikdash by establishing a synagogue.The Shulhan Aruch’s ruling applies also to ensuring that there is sufficient capacity in the town’s synagogue(s) and building more if necessary.
Rabbi David ben Zimra (Shu”t HaRadbaz, vol. 3, § 472) discusses the propriety of building a new synagogue because some congregants wish to pray using their own rite. He explains that one must pray where one’s heart desires. In such a place–where one sings with familiar tunes and with practices to which one is accustomed–one’s mind gains full composure and one’s soul is awakened, and therefore one’s prayers are most effective. Thus, it is permissible to establish a new synagogue so as to conform to the customs and rites of a portion of the congregants. Nevertheless, Rabbi Shalom Schwadron (Shu”t Maharsham, vol. 3, § 168) rules that if by splitting off it will significantly affect the membership of the original synagogue such that it may not have a consistent Minyan or worse, may close, then it is forbidden.
Summary: Every Jewish settlement must have a Synagogue and a Bet Midrash. A new synagogue may be established as long as it does not affect the membership of the other synagogue(s).