The Shulhan Aruch rules that starting from the second night of Hanukah, when multiple candles are lit, the kindling should be done from left to right. This means that one first lights the left-most candles, then the one to its immediate right, and so on. In light of this Halacha, the Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) writes that one should position himself to the left of the candles, so that the first candle his hand encounters is the candle that is to be lit first. The Halacha of "En Ma’abirin Al Ha’misvot" establishes that one should not pass over an opportunity to perform a Misva, and thus one should not pass over the right candles on his way to light the left-most candle. Hence, it is proper to stand to the left so he first encounters the left-most candle.
It is customary on Hanukah to light candles each night in the synagogue, and to recite Berachot over the lighting. If a synagogue has several Minyanim for Arbit, it is preferable to provide enough oil to sustain the candles until after the later Minyanim. If the candles are indeed still burning when the later Minyan begins, the people do not have to blow out the candles so they could light again. And if the candles blow out before the later Minyan, the people should relight the candles, but the Berachot should not be recited. As it is, the recitation of Berachot over the synagogue candle lighting is somewhat surprising, as this lighting is not Halachically required and is rather done by force of custom, and we do not normally recite a Beracha over a custom. The Poskim provide different explanations for why Berachot are nevertheless recited, but as this itself is surprising, we should not go even further to require reciting Berachot at each Minyan. Additionally, although some authorities maintain that every Minyan must light Hanukah candles, others are of the view that the obligation is upon the synagogue as a whole, and not each Minyan. In light of this question, too, Berachot should not be recited when the candles are lit again at the later Minyan. This is the ruling of Hacham Bension Abba Shaul (Israel, 1923-1998), as it appears in the work Ner Sion (p. 333).
Summary: Before lighting the Hanukah candles, one should stand to the left of the candles so that he first encounters the left most candle, which is to be lit first. If there is more than one Minyan in the synagogue for Arbit, and the candles lit before the first Minyan are still burning when the second Minyan prays, the people do not have to extinguish the candles and light them again. If the candles have burned out, they should be lit again, but the Berachot should not be recited.