It would depend very much on the republic that the Soviet citizen was born in and living in, as they all implemented different citizenship laws, sometimes strikingly so.

At one end of the spectrum would be the RSFSR (Russian Federation after 1992). Until 2000 or so, Russian citizenship law stated that any Soviet citizen residing in the Russian Federation was eligible for Russian citizenship. Russia also allows dual citizenship, so in theory someone born in a different SSR could hold Russian citizenship and their native citizenship.

In practice it doesn't really work this way, as pretty much every other former SSR bars holding multiple citizenships under their laws. The exceptions can pretty clearly demonstrate the rationale why: the breakaway Georgian regions of Abkhazia and South Ossetia, which are not widely recognized, allow dual citizenship status, and almost all of their citizens (90%+) hold both the local citizenship and Russian citizenship to the point that they are de facto parts of Russia.

As far as the other end of the spectrum, Estonia and Latvia are the prime examples. In their case, at independence they only extended citizenship to residents who either were citizens of the former independent states of Estonia and Latvia respectively in 1940 (at the time of Soviet occupation and annexation), or their direct family members. Anyone else had to apply for citizenship, including taking extensive language tests. The result of this was to make ethnic Estonians and Latvians citizens of the newly independent republics, while forcing the large Russian-speaking populations (25-30% of the countries' populations) that had largely immigrated there after the Second World War to be effectively stateless. The stateless populations have decreased (and have been extended a number of rights short of full citizenship), but are still sizeable.

The UNHCR (United Nationa High Commissioner on Refugees) has a number of reports about citizenship status in parts of the former USSR.

This report from 1993 runs through the new nationality laws for each former Soviet state. The main takewaway:

In brief, in their nationality laws, Estonia has opted and Latvia is opting for the restored-state model. Lithuania and Moldova have followed a mixed system and have determined their initial body of citizens partly with reference to the situation prior to the Soviet annexation and partly to residence in the newly independent country. Other former USSR Republics have drafted their nationality laws adopting the new-state model and determined their initial citizenry on the basis of permanent or prolonged residence, or simply, residence on their territory on the day of entry into force of the nationality law. Azerbaijan, finally, chose quite an unusual way and based its initial citizenry on the situation previously existing in the Soviet Socialist Republic of Azerbaijan before it attained independence.

It's also worth noting that despite these laws, outside of Estonia and Latvia there are still non-trivial numbers of stateless persons, ie people who hold no citizenship. The UNHCR works with republics to solve this issue, and a recent report on the status of stateless persons in Central Asia is here. The report also notes that even in the Soviet period, immigrants or international students could be registered as stateless residents of the USSR, so the Soviet Union had a stateless population even before the breakup.

Sources:

UN High Commissioner for Refugees (UNHCR), Nationality Laws of the Former Soviet Republics, 1 July 1993

UN High Commissioner for Refugees (UNHCR), Statelessness in Central Asia, May 2011