Initially, Stalin wanted to bring National minorities (Georgians, Ukrainians, etc) into the Russian Soviet Federated Republic and make them citizens/subjects, but Lenin took his idea and modified it into what would become the USSR. Each Nationality or "group" would have their own Soviet Republic and have some Autonomy over their day to day affairs, but be ultimately answerable to the Soviet Union.

Lenin and Stalin's ideology was that "Nationalism" was a bourgeois concept, destined to fade away in the wake of their Communist revolution. In practice, it wasn't that easy. The many peoples under the Czar - Russians, Georgians, Ukrainians, Etc - were too diverse and too different to unify under a single system. There were also Various revolts and uprisings against the Soviets, for example, in Ukraine. Lenin and Stalin decided to be pragmatic and to give the various Nationalities in the Soviet Union some form of Autonomy. Especially after the 1920s Ukrainian Famine, Lenin saw the importance of giving certain people what they wanted to make sure that the Soviet Union worked properly. So, he decided to placate Ukraine by giving them limited opportunity and permitting them to develop and preserve their own national identity/culture. Stalin would later reverse this and use the Ukrainian Famine/Holodomor as an opportunity to purge Ukraine of "Nationalists", and make it more subservient to the USSR.

To take Ukraine as an example, being an important part of the USSR (main food supplier), Ukraine had its own Government, its own Communist Party, and its own Police Force, but was ultimately answerable to the USSR and Stalin's government. The Ukraine Cheka nominally answered to the "Main" Cheka in the USSR itself, but was directly under the control of Stalin, because the Police Chiefs - especially Stanislas Redens (His brother in law) and Vsevolod Balitsky - were hand picked by Stalin and known for their loyalty to him. When the 1932-33 famine/Holodomor happened, it was initially the Ukrainian Soviet Republic that was responsible for running the affairs of Ukraine and responding to the Famine, until Stalin and his Emissaries stepped in.

The Government of Ukraine, much like the USSR, was headed by the Council of People's commissars (Cabinet), which was headed by Vlas Chubar during the 32/33 Famine. The Government in turn was answerable to the Ukrainian All Union Communist Party (Bolshevik party), headed by the Central committee. This was headed by the First Secretary, Stanislas Koisior, who was handpicked by Stalin. Because Ukraine played a vital role in the Soviet Economy, Kosior was also a member of the Politburo, the Highest Governing body of the USSR, along with Stalin and other high officials. When Stalin stepped in WRT to the Famine, the first thing he did was appoint Pavel Postyshev - another loyalist - as the head of the Kharkiv Bolshevik party. At the time, Kharkiv was the Capital of Ukraine, which made Postyshev - and by extension Stalin - Kosior's direct supervisor.

The same was true for other Soviet republics. The Local Government was modelled after the USSR's, and were answerable to the Communist Party, who were in turn answerable to the Communist Party of the USSR.

In summary, they were free to run their day to day affairs but were ultimately answerable to the Soviet Union. Lenin's actions can be taken as a general example. A place like Ukraine was too valuable to let go/lose, so he allowed Ukraine to have a nominally independent government and preserve their national identity, while still providing for and being answerable to the USSR.

Reference

Stalin, new biography of a dictator, by Oleg Khlevniuk.