I can only comment on Estonia, since that's the only republic I have any academic sources to hand on, but I don't get the sense that Estonia enjoyed a particularly higher or lower level of freedom than the other republics.

It's important to acknowledge that, legal separation of powers aside, the RSFSR accounted for about 80% of the USSR's economy and population and about 90% of its institutions of higher education, so the reality was that Russia was going to dominate even if the republics had been granted substantive economy. Its not parallel to the USA, where there's no single state that dominates.

Anyway, on to specifics. Constitutionally and legally, the subject republics were granted very substantial autonomy and a say in central government proportional to their population. If an alien had come to earth and read the various constitutions he'd have concluded the USSR was a very decentralised state.

In practice, the authorities in Moscow kept an extremely tight grip on everything that was deemed 'strategic', which they defined extremely broadly. In Estonia, factories that produced defense components or raw materials used in defense or other nationally critical industries (for example components in nuclear power plants) were essentially totally outside the control of the Estonian government, whatever the legal reality. The same is true of all localities close to major military bases, which were directly managed by the armed forces, who responded to Moscow, or areas close to international borders or international waters. The republics also had no say in the management of the troops based in their territories - the Estonian government didn't even officially know how many troops were based in the country.

Areas that the subject republics did have autonomy in included: Language and cultural policies (They were able to produce films, books, newspapers etc in their local language as well as Russian), education policies (Many Jewish academics relocated to Estonia, and Estonian schools had eleven years of secondary education, unlike the rest of the USSR where they had ten).

So, in summary, the republics had some power to make decisions in non-strategic areas, but they did not have power comparable to a US state.

Source: Toivo U Raun, "Estonia and the Estonians"