The following was written in a message by Rivka Schirman:
1. Kehilot were autonomous bodies all over the Polish-Lithuanian commonwealth and were abolished by the occupying forces in their respective territories in 1792.
2. Kehilot in Galicia were re-instituted by the Austrians to their original Polish-Lithuanian function when Galicia was under Polish autonomy.
3. The Kehilot were granted autonomy in the Second Polish Republic by the Constitution of 1921 nationwide, but this was not applied in formerly Russian occupied Poland until the Ugoda agreement of 1924. In Galicia, things simply continued as they have always been since Polish autonomy was granted to Galicia in 1875. In a way, one might say that the Ugoda made formerly-Russian-Occupied Poland align itself along the east Galician (i.e., the Historical Poland) lines of Polish-Jewish relations.
4. The autonomy of the Kehilot refers to the elections of their bodies (as opposed to the formerly Russian system where a or some representative was appointed by the governor and not a body elected by the community); they were, as all religious bodies in Poland, including Catholic bodies, to be auto-financed and not government financed.
5. Kehilot never ceased to exist until WWII -- internal dissensions did exist and were apparent in each election to this democratically elected autonomous body but this is nothing new -- you take two Jews and you have three opposing political Jewish parties. Financial difficulties were both a result of these internal political dissensions (how
should the money be distributed and for what purposes, and some of the tax-payers refusing to pay their dues) and of objective difficulties (some of the tax-payers being unable to pay, general economic crisis). In case the financial crisis was too big, either the Kehila appealed to the US Landsmanschaft for donations or (rarely) it ceased its autonomous existence and became a branch of a larger neighbouring Kehila.
6. As for the specific function of the "official or authorised rabbi" I have not researched this specific issue, but I would presume, from the research I have done that it had a lot to do with (a) the overall financial situation of each individual Kehila and, mainly (b) the need of the Kehila for such rabbi. I presume that in Kehilot where the majority belonged and voted Agudat Yisrael, for example, there was no need for a budget line for a local Polish speaking rabbi, and those who absolutely wanted to officially register their marriage went to the departmental or regional one. It might be interesting to compare the results of the internal elections of the Kehilot (details are given in Pinkas Hakehilot) and try and compare the results of these elections to the number of vital records where children bear the name of the mother, i.e., linking the result of the internal elections to the apparent absence of local "authorised" rabbi. But this is a full PhD in itself andI don't have time (nor energy) for it right now.