In the Second Temple period, who ruled the synagogues? It seems the general scholarly consensus is that synagogues, especially those of the Diaspora, belonged to the lay people. Two scholars offer a different picture.
1. Donald Binder, in his book Into the Temple Courts, suggests the priests and Levites controlled some (many?) of the synagogues. The Greek terms usually used to denote synagogue leaders in literary and epigraphic evidences include archon (18 times: 2 in Jos.; 16 in epigraphs) and archisynagogos (16 times: 10 in NT; 6 in epigraphs), among several others (which occur between 1 and 4 times). Archon ("prince" or "ruler") appears in the form of archontes in the LXX in Ex16.22; 34.31; Num1.16; 31.13,26; 32.2; Josh9.15,16; 22.30. "During the Second Temple period, these archontes, along with a somewhat exclusive group known as 'elders' (presbyteroi), served as subsidiary rulers under the High Priest, forming a ...synedrion" (Temple Courts, 345). Binder then goes on to show that these archontes also run the synagogues of both Judea and the Diaspora. Entry into the synedrion "was probably by the appointment of the High Priest, though the lineage and popularity among the masses were undoubtedly factors as well" (345). Binder also notes that "terms appear to have been for life, though changes in political regimes could result in the loss of one's position" (345). Further, Binder demonstrates that doorkeepers, those conveying scriptures to and from readers, and general overseers of synagogues (at least in Egypt and Palestine) were often Levites.
After his examination of all the data, Binder concludes, "Our survey has highlighted the role of priests and Levites within synagogues. Here, sources indicate that priests served as archisynagogoi and archontes, and suggested that they frequently served as scribes. Similarly, the Levites functioned as scribes as also appear to have filled the role of synagogue attendant... The evidence points to the conclusion that the Temple and the synagogue both belonged to the priests, Levites and people, with all three groups having a measure of leadership and participation within each institution" (371).
2. E. P. Sanders, in his book Judaism: Practice and Belief (63BCE-66CE), makes the same claim, though on different grounds:
"Philo indicates that priests retained their status as leaders in the Diaspora (Hypothetica 7.12f.), and archeology confirms that in at least some places outside of Palestine priests were specifically designated as such [footnoting an inscription found at a synagogue at Sardis, dating from the 4th century CE, reading "priest and teacher of wisdom"]" (pp. 52-3).
"[Upon assessing the well-known Theodotus inscription] What is clear here is that the rulers of the synagogue were priests, three generations of them, and very prosperous priests at that. If we must assign them to a party, the Sadducean would be the most likely guess [contra Hengel, who believes Theodotus was a Pharisee], but there is no reason to think that they represent a party. What we learn from the inscription is that a family of wealthy priests who could speak Greek built and maintained a synagogue for Greek-speaking pilgrims, and that the synagogue had a dual purpose of serving as a guest house and a place of instruction. The inscription supports the evidence of the literature: it was the priests who taught the law" (pp.176-7).
"The Theodotus inscription is graphic evidence of the role of priests in synagogues, a role that some retained in the Diaspora. We recall that according to Philo a priest or elder was responsible for sabbath instruction (Hypothetica 7.13). At the synagogue in Sardis an inscription was found that refers to a man who was a 'priest and teacher of widsom'. This is from the fourth century. Its relevance is that it shows continuity with the passage of Philo and the Theopdotus inscription. In neither Palestine nor the Diaspora did priests withdraw from public life and community study and worship. By our period, prayer and reading of the Bible had already been incorporated in the temple service. It was a natural development for priests to perform both functions in synagogues as well.... The priest or elder read and interpreted the Bible, and other for the most part remained silent (Philo, Hypothetica, 7.13)" (pp.201-2).
"...Josephus presupposes that the priests were the official teachers of the nation, though he also depicts lay Pharisees and Essenes as public teachers. I think that we cannot safely generalize about who dominated how many sunagogues, but we must doubt that the Pharisees ran all of them" (p.398).
On page 398, Sanders sufficiently shows that there were three times more Levites and priests than Pharisees, based on Josephus' ficgures. And that the number of Pharisees made it impossible for them to be in charge of all of the synagogues in the Diaspora. Thus, he concludes, the priests and Levites were in charge of some of the synagogues.