You would be correct that the meme of 'quantity has a quality all its own' is not representative of reality. Broadly speaking, I think you could expect a Soviet Army rifleman to have received a similar amount of training time as their counterparts in the British Army of the Rhine, US Army V Corps, et cetera. I choose my wording carefully, because this does not mean the two sides were similarly trained - the Soviet Army was not a mirror image of the US Army. The Soviet Army was a conscript army, of course, though this was also the case in most of the Western armies throughout the Cold War. The British phased out conscription in 1963, the Americans in 1973, and the Dutch retained it until after the end of the Cold War.

There are three major competing perspectives on this question, I think, and they're all sort of true.

1. On one side are Western assessments that Soviet troops were well-drilled and could generally be counted on to perform well in combat, but the nature of the Soviet military system produced junior officers and NCOs who lacked initiative and flexibility.
2. The second perspective holds that, particularly in the late Cold War, the Soviet Army emphasized flexibility and initiative more than it gets credit for. This perspective implies that any given platoon of Soviet infantrymen would perform comparably to a western platoon, though not identically.
3. The third perspective, and this is one that you see among both western and Russian sources, is that despite the seriousness of the Soviet military as a whole, the level of discipline and morale among the lower ranks was quite poor. You start to see this perspective a lot during the Afghan years, and then the poor performance of the Russian Army in Chechnya sort of cemented this idea.

As I said at the beginning, these perspectives are not mutually exclusive. The conscription system meant that most Soviet non-commissioned officers were simply high-performing conscripts, selected for a sergeant's billet during initial training. This was in contrast to the Anglo-American system, in which long-serving NCOs were a font of discipline and institutional wisdom for the enlisted men. Inexperienced NCOs would have had a hard time passing on good habits and information to their juniors, as they of course did not have that experience themselves. Partly as a result, Soviet training did emphasize performing a variety of drills to certain norms of performance: a rifleman must be able to land five rounds on such-and-such target size at 300m from a standing position in eight seconds, your squad must accurately execute a Reaction to Contact drill in less than 45 seconds, etc etc. As Christopher Donnelly summarizes in The Soviet Military: Political Education, Training and Morale,^(1)

The Soviet military system requires the conscript to perform simple battlefield tasks which have been learnt by constantly repeated drill, but to be able to perform them in any battlefield conditions.

It is Soviet experience that reasoned thought is the first victim of battlefield stress and the Soviet military system does not therefore expect initiative or imaginative and constructive actions from its soldiers: it only expects reaction and drill movement. The very concept of initiative (in as much as it is ever asked of NCOs and junior officers) is taken to be no more than the initiation of the correct drill appropriate to the given circumstances.

Once he has completed his initial basic training the Soviet soldier is tested regularly on all the skills which he is expected to acquire. In training and on exercise, every action which the soldier or NCO and officer does is measured against an average standard or norm... As the training tests are done against a standard or norm, at the end of each month the progress of the soldier's training can be assessed with reasonable accuracy.

However, there is documentary evidence that this is not the whole picture. For instance, a 1980 training manual for noncommissioned officers tells us that all soldiers and NCOs must "show courage, initiative and resourcefulness in combat," and that it is the responsibility of NCOs specifically to "be an example for subordinates of initiative and courage."^(2) There are also plentiful recorded descriptions of junior-officer and enlisted initiative from Afghanistan; Lester Grau's The Bear Went Over The Mountain contains several vignettes of platoon-level tactical actions in which lieutenants and enlisted men display independent military thinking similar to what we might expect to see in a modern ISAF or MNF-Iraq platoon-level after-action report.^(3) The Red Army's experience in World War II is also full of stories - some probably Stakhanovite propaganda, some true - of enlisted men's initiative. Wartime manuals on reconnaissance and maskirovka (deception) practices emphasize the cunning and individual initiative of soldiers as well (see this older thread for more detail, ft. /u/hamiltonkg) The Soviets took every possible lesson they could find from World War II and tried to apply it forward, a practice which has even persisted in the current Russian Federation's military journals, and although I have not seen any articles in Voenny-istoricheskii Zhurnal or Voenny Mysl' which discuss soldier initiative, I would lean towards assuming that these lessons were not forgotten.

The third angle, discipline and morale, relates largely to poor barracks living conditions and the brutal, almost ritualistic hazing known as dedovshchina (lit. "grandfathers' rule"). Dedovshchina is the customary hazing of younger conscripts by those who are already on their second contract year. In units with better discipline, it might 'only' take the form of junior conscripts being forced to clean for their seniors, for example. In worst-case units, there are credible reports of beatings, gang rapes, and murder. Dale Herspring's 2007 Journal of Slavic/Soviet Military Studies article "Dedovshchina in the Russian Army: The Problem That Won't Go Away" covers it well, I believe. Dedovshchina was a cause for desertions, self-inflicted wounds, suicides, and drug use in Afghanistan in particular, and then in the grim post-USSR decade as well. As with American units in Vietnam, the nature of fighting long 'savage wars of peace' with conscript armies seems to have exacerbated discipline problems - especially dedovshchina - in 40th Army during the occupation, to the point that it features heavily in popular narratives of the war. Whether this would have played a significant role in Soviet morale during a 'hot war' with NATO in Europe is unknowable, though as far as I am aware dedovshchina was less of a problem in the Group of Soviet Forces in Germany and other 'first-line' units which would have seen initial combat in that scenario. As Donnelly notes in one of his chapters in the aforementioned book, in general it seems that soldier quality of life has to get really bad before it seriously degrades combat performance in a knock-down drag-out high-intensity war, whereas small morale problems become magnified in occupation and counter-insurgency conflicts.

Notes and Sources

1. E.S. Williams, C.N. Donnelly, and J.E. Moore. The Soviet Military: Political Education, Training and Morale (New York: Palgrave Macmillan, 1986) 109-110. Although this is a Cold War work and contains some excess political commentary on the inherent dehumanization of communism, Donnelly was one of the founders of the Soviet Studies Research Centre at the Royal Military Academy Sandhurst, which produced some really solid work on the past and contemporary Soviet military. SSRC and the US Army's equivalent (Soviet Army Studies Office) were at the forefront of the nascent revolution in English-language Soviet studies in the 80s.
2. Учебник сержанта мотострелковых подразделений [A Textbook for Sergeants of Motor Rifle Units] (Moscow: Voenizdat, 1980) 4-5.
3. Lester Grau, The Bear Went Over the Mountain: Soviet Combat Tactics in Afghanistan (Washington D.C: National Defense University Press, 1996) See for instance p.167, "An airborne platoon conducts an ambush in Helmand Province."