General Division of Responsibilities
The division of responsibilities was roughly similar to that at the national level: the decisions were made at the party committee (and both Soviet chairman and executive committee chairman were members thereof, together with the police and KGB bosses) under the guidance of the secretary, and the Soviet chairman was responsible for getting the Soviet to rubber stamp the decisions and the executive was responsible for, you guessed it, executing the decisions.
The 1st Party secretary was the supreme boss in the Russian and Ukrainian regions (oblast/krai). In the ethnic regions, the 1st secretary was "the local", representing the ethnic/tribal elites, and the 2nd secretary was a Russian (or, rarely, Ukrainian or even Belorussian) sent in from the Center, and it was the 2nd secretary who (unofficially!) held the supreme power (but not the official prestige and visibility!), because the power in the USSR flowed from Moscow to regional centers, and from those to local centers &c.
The only exception I can recall was Kolbin whose appointment as the 1st Secretary in Kazakhstan in December 1986 lead to riots.
The possible exceptions in the other direction were Aliyev (who had KGB roots and thus did not need a Moscow baby-sitter) and Kunaev who was a personal ally of Brezhnev.
Post-USSR (response to a comment)
When the USSR collapsed in 1991, the 2nd secretaries lost their power source - Moscow was no longer to be reckoned with - and the 1st secretaries landed with all the central power there was.
Source: I lived there and observed the politics first hand.