Let's split up the odds of hitting the target into two criteria: mechanical precision of the weapon being used and the ability of the crew to effectively exercise it. I'm going to take the 122 mm D-25T gun mounted on an IS-2 tank as an example since that's the gun I have the most data on.

Let's start with the former: with a perfectly trained crew that has mastered their weapon completely, will the gun be able to hit a tank sized target at "long range"? The maximum range at which gunnery trials took place was usually 2000 meters (or 2000 yards for Imperial measurement system users), so let's settle at that.

Practical trials show that at a known range 100% of shells fired from an IS-2 tank at a range of 2000 meters will hit on a vertical target within a circle with a radius of 120-130 cm. The 50% of the shells closest to the center will fall in a circle with a radius of 45-90 cm. Given that a tank sized target is about 3 by 3 meters, there is no way an IS-2 can miss at 2000 meters in ideal conditions. Note that the data demonstrates the precision of the gun rather than the accuracy, but provided that our skillful crew calibrated their sights correctly that should not be an issue. This also assumes that the target is completely visible and not moving.

Now we move on to an issue that might upset this ideal scenario: range estimation. If the crew knows that the enemy is exactly 2 km away they can just aim according to their ballistics table and fire away, but the enemy has a nasty habit of not following rules. Let's see if these ballistics tables can help us figure out what happens if there is error.

Conveniently, the tables give a point-blank range for our gun. What this means is the range at which the trajectory of the shell does not exceed the height of your target, meaning that if you fire with your sights set at that range and the target is actually closer, you will not overshoot. The height of the trajectory at 2000 meters is 9.1 meters, so there is a very real chance of overshooting our enemy tank if we guess wrong, but the height will not exceed 3 meters up to a range of 1180 m, so even a relatively inexperienced crew will have no issue making a hit at that range. This is also assuming that you and your target are on level ground, if you are higher or lower than they are you will also need to make corrections to your sights. All of these tables were calculated at +15 degrees C at 750 mmHg, so if your temperature or pressure differs, you have to make corrections too. Barrel wear also requires you to make corrections. Wind? You better believe there will be corrections. Our perfect tank crew will have to take all of these things into account to make their first shot hit. More realistically, they will use a technique called bracketing: you fire two shots, one to the left of your target and one to the right. Now they have two reference points to work with and a skilled crew will almost always land the third hit.

So far we've been talking about a completely passive opponent standing out in the open, but as you mentioned, tanks tend to move around, or at the very least attempt to present as little of themselves to the enemy as possible. For a moving target you will have to predict where it will be when you fire. Our shell fired from the D-25T will hit its target in 3 seconds at 2000 meters or 1.3 seconds at 2000 meters. Assuming the enemy tank is travelling at about 30 kph, that's a distance of 10-20 meters we will need to offset our sights by, quite a significant adjustment. Note that unless the enemy is completely unaware of you, the gunner will only have several seconds to estimate the speed and direction of the tank, enter the corrections, and fire. The enemy driver will either attempt to duck behind terrain as soon as possible or move in an unpredictable zig-zag motion to avoid being hit.

As for hull down, that's a different story. I don't have Soviet data on this subject, only some British data. The findings are pretty dire though: they estimate that while firing 17-pdr APCBC at the side of a Tiger results in a 30% chance of success at a range of about 3000 yards (note that this means a hit and a perforation of the armour), 50% chance of success at just over 2000 yards, and 90% chance of success at under 1000 yards. For a hull down tank these odds drop drastically: 30% chance at just over 1000 yards, 50% chance at under 1000 yards, and the marking for 90% chance of success does not appear on the diagram at all.

As you can see, there are a lot of factors that go into whether or not a crew will land their shot, both their own skill, the characteristics of their weapon, and what the enemy is doing at the time. Once again, I don't have data for the D-25T, but looking at a summary of hits taken from 75 and 88 mm guns over the course of the war you can see that these extreme ranges of engagement are very rare. According to Soviet operational research, less than 11% of hits from 75 mm guns were scored at a range of over 1000 meters and 19.3% of hits from 88 mm guns. Western Allied operational research agrees with these findings: even in the vast open battlefield of North Africa the average engagement range was 900 yards, the more varied battlefields of North-Western Europe had this range drop to 800 yards.

These findings more or less agree with the data shown for the Soviet 122 mm gun. While it is definitely possible to hit a tank-sized target at a range of well over 1000 meters, you are going to have a tough time of it outside of ideal conditions: clear sight of the target at a known range.

Sources:

http://www.tankarchives.ca/2013/02/accuracy.html

https://www.tankarchives.ca/2020/06/bullseye.html

Tablitsy strelby 122-mm tankovoy pushki obr. 1943 g D-25T TS-#144T

http://www.tankarchives.ca/2013/08/combat-performance-of-75-cm-and-88-cm.html

https://earlesspierre.livejournal.com/88854.html

Survey of Allied Tank Casualties in World War II ORO-T-117