The truth is that the KV-1 and the IS-2 were superb tanks for the time but they had obvious drawbacks. Your question is about how big or how small a role they had on the Eastern Front.
Well, frankly, the only real answer to this is that they didn't have much of a role at all, mostly because the Soviets had the T-34 and T-34-85, which did most of the lifting in the East, both of the tanks you named never got the needed numbers to play more than supporting roles. For several reasons, lets start with the KV-1. The KV-1 was a very tough tank, it outclassed all of the German tanks and guns at the outset of Operation Barbarossa, with the exception of much heavier, rarer guns. So, why did they play a small role? The T-34 was almost just as hard to knock out than the KV-1, it was cheaper to produce, in both time and raw materials, it was faster and was far more versatile. So, the USSR ended up building around 500 KV-1s and they were mostly produced before the start of fighting on the Eastern Front.
Commanders loved the idea of the KV-1 but they were so clunky and so hard to use that the soldiers hated them in comparison to the T-34, especially gunners and loaders. Who had to contend with very poor visibility and ammunition access because of the atrocious turret design. They had a very widely recognized problem with their clutches, drivers would often report that the clutches wouldn't move and would need to be handled by hitting it with a hammer... This gives us a very important clue as to just how clunky these things were. There is evidence that the KV1 stayed in production as long as it did because the Soviet soldiers were very scared of reporting problems with the tank. The stories of the time state that if you wanted to have a hope to not get shot behind a shed by a political commissar, you'd need to be a recognized Hero of the Soviet Union to report a problem. So, it stayed around, it was afterall, a rolling bunker in essence, so it had its uses but the Soviets would have been far better served building more T-34s rather than building KV-1s.
As for comparisons with the Tiger I and II, well, this will be short, the KV-1 was completely outclassed by both of those tanks, they simply weren't comparable, they were worlds apart. By the time the Tigers come around the KV-1s have mostly been replaced or are being replaced with other variants of KVs.
The IS-2 is a much better comparison with the Tigers. The IS-2 outclassed the Panthers easily, it had a better gun, speed and armor. The Tiger I made for an interesting opponent in many respects, the IS-2 had better armor and a better gun than the Tiger I. The gun was better because while the two guns had similar AP properties, the IS-2 had far superior HE capabilities. The IS-2 had better armor, with the Tiger I being able to pierce the frontal armor of the IS-2 at around 600-700 meters while the IS-2 could open up the frontal armor of the Tiger I at around 1000 meters. Since the IS-2 had superior armor, it had a better time against the Tiger I at around 1500 meters, where the Tiger I needed better hits to knock it out. The Tiger I was also heavier and slower than the IS-2 but it did have better optics and far superior rates of fire.
The Germans did not employ a high number of Tiger IIs in the East but the Soviets did encounter them and when they did, the IS-2 proved to be superior to the Tiger II. Why? Because the Germans at the end of the war were finding it harder and harder to produce quality alloys for the welding joints of tank armor and the Tiger II had a lot of them. The Soviets found out that most of their tank guns could shake free the front plates of the Tiger II from the welds if you just hit it twice in a row. The IS-2 was much more mobile and did not have the grave reliability problems of the Tiger II, which was prone to overheating because it had faulty gaskets and seals, which Germany had a hard time producing in the late war, and its drivetrain was meant for much lighter vehicles which made it overburdened... All of which contributed to the sometimes poor performance of the Tiger II because the lack of training the characterises late-war German tank crews amplifies any difficulties that the tank had with maneuverability. Its gun was superior in all aspects to the IS-2s except again for its HE qualities and the armor, when the welds did not give way, was superior as well, though not penetration proof. So, with all the disadvantages I've put down about the Tiger II, why is it so mythical? Its gun, its gun, its gun. It was a beast. It could open up anything the Allies or Soviets could throw at it. Including the IS-2. It could in fact penetrate the frontal plate of an IS-2, which nothing else could do during the war. So, from what we know about the Tiger II, it would have been a very difficult match-up for any of the belligerents in WW2 if it had come up sooner, before the Germans started having industrial material problems or crew training deficiencies.
Hope that answers your questions.
Sources for the Tiger II: You'll have to forgive me but I couldn't find all of them in preview mode. Here they are nonetheless... Source
The heavy Soviet tanks had a different purpose than the Tiger 1 and were mainly used in static defence or as an offensive breakthrough tank. After the bunkers and entrenchments were taken on by infantry units supported by the heavy tanks, the T-34 would take over to exploit the gap in the lines. Because of this different tactical deployment the Tiger 1 was most likely encountered by a T-34 rather than a KV-1 or IS-2. Also when the KV-1 was in use, Germany hadn't developed the Tiger yet and the IS-2 only became operational during the very last phase of the war. The Tiger 2 was more or less similar to the IS-2 and they met in battle during the Battle of Berlin.
You will find that the Russians had quite good tanks. The T-34 was an excellent tank due to its wide tracks enabling it to be used effectively even in the snow and subsequent snow-melt where the Panzers suffered heavily. Its amarament, the 76.2mm high velocity gun was known as the best tank gun in the world at the time.
That being said in a 1v1 confrontation the Tigers would wreck, the T-34's shots would simply bounce of the ridiculusly thick armour on the Tigers. However Tigers had slow acceleration, could be felt and heard from kilometers away, were very fuel inefficent and very inmmaneuverable; you would never confront one from the front, you would ambush them from the sides and or behind and take out the tracks, fuel tanks or ammunition stores.
My opinion is that the Panzers are heavily overated tanks. They really never became the massive, impenetrable behemoths that we imagine they are until the advent of the Panthers and Tigers and by that time it was already too late for them to be used to their full extent.
I have a source dont worry the Book is called World War 2 but ill have to edit this post with the authors name as I dont have it on me and the title is a bit broad to search for.
Found it: The authors name is Antony Beever and the title is The Second World War. http://www.amazon.co.uk/Second-World-War-Antony-Beevor/dp/0753828243
KV1 was almost immune to PZIII and PZIV tanks. Many times the only way to destroy it was a point blank shot in the rear. But when T34 was introduced, it took the place of main battle tank. There was no need to use KV1 since T34 outclassed it in almost every single way. Later on KV series used as a base to the IS tanks. But by the time those were used (IS2) USSR and allied forces were deep in german territory.
TL;DR KV was outmatched by Tigers. IS2 could take an equal fight with Tigers. But KV was worse than T34 and IS came too late. So imho they had a small role
The kv 1 was the original heavy tank of the soviet army. That being said, the kv1 was not very good as its only advantage over t34 was thicker armor. The gun was identical and mobility was worse. The is2 was symbolic at least since it bore joseph stalins name. It was also probably the most powerful tank of the war with massive armor and a 122mm gun that outclassed any other tank gun at the time. An impact from a he shell could do significant concussion force damage to a tiger whilst the ap round was known to punch through the frontal armor of a tiger. Although not deployed in the same numbers as the t34 it was at the forefront of the soviet counter offensive