The practice of constructing nuclear warheads seems like one that would be well hidden for a number of reasons. Since both countries were in an arms race in the second half of the 20th century, what was to stop either of them from lying about the numbers of stockpiled weapons? How could two countries with major distrust come to terms on the very thing driving their competition, assuming they ever did?
The idea of stockpiling these warheads has also seemed a bit silly to me. The line I recall being fed in history class was that the USA was trying to drive the USSR into financial ruin, but that seems like a simplistic view of what drove the motivation for a surplus of explosives. I’m deviating from my original question here, but was there a more practical reason for the USA creating warheads that would ultimately be excessive? You can’t blow up the world more than once.
First, it seems like your question is based on a misconception that the US and USSR declared their warhead stockpiles during the Cold War. They didn't. Both countries considered these state secrets. There were estimates of these warhead numbers produced by both government and non-governmental sources, but the arms race did not revolve around official statements about warhead counts. At times the US and USSR both over-estimated their opponents' (they rarely underestimated) warhead counts. The US, in the post-Cold War, declared its historical warhead stockpile size, but other than a few years during the Obama administration it has not declared its current warhead stockpile size.
Now, if you are talking about treaties, it is worth noting that none of the existing treaties regulate the number of stockpiled warheads. This is because, as you note, verifying these numbers is difficult. Warhead designs are still (for better or worse) kept secret from other nations. Confirming that a given warehouse contains warheads and not warhead-shaped decoys is very difficult to do without intrusive inspections. (But not impossible! But that's a different story. There are various efforts being made to talk about how you'd count warheads if that was a thing you wanted to do — e.g. if you had a treaty that would severely limit them — but this is all still in the "let's study the problem just in case we need to deal with it in the future" stage of things at the moment.)
Instead, treaties count things that are easy to count: submarines (which have to dock periodically and so can be counted), bombers (which have to sit on runways), silos (which are easy to spot), and so on. If I say, "I will only field X number of missiles" — that's relatively easy to confirm! I can count the missile silos on your subs and count the missile silo holes and determine whether you're telling the truth. Want to convince me that you got rid of a missile silo? Take it apart and fill it in with cement and I'll watch from afar and be satisfied. Want to convince me you got rid of some bombers? Chop off their wings with a giant guillotine while I watch with a satellite.
With each of these treaties the question of verification — how you detect and deter cheating — has been central. So what is monitored tends to be directed by that consideration as well. That being said, there have always been accusations of cheating and a lack of perfect fidelity with such things. And some trust is needed; total paranoia won't lead to treaties. But the verification measures are designed to build confidence and keep things within certain limits.
Anyway the question about why the stockpile sizes got excessive is a complicated one. There is some discussion here and here. The answer is a mixture of historical forces, technical constraints, ideological aspects, domestic and international politics, and so on. It is not a simple thing (and the answer is not "the US was trying to spend the USSR into ruin" — that is an after-the-fact justification).
The short answer is that neither country trusted the other.
Effectively every treaty limiting the number of nukes came with all kinds of ways to try and ensure nobody was cheating.
To look at a few.
INF (Intermediate Range Nuclear Forces Treaty (1987) included provisions for 5 different kinds of on site inspections, sharing of telemetry data, and other cooperative measures.
SORT (Strategic Offensive Reduction Treaty (2002) also includes multiple types of on-site inspections, sharing of telemetry data, and cooperative measures.
START 1 (strategic arms reduction treaty) (1991) Includes 12 different types of on-site inspections, provisions protecting the use of satalites to gather information for enforcement, sharing of telemetry data.
Basically every other nuclear treaty has included similar provisions, except for a few that where specifically about sharing nuclear technology with certain countries, and those treaties normally between allies.
Now I want to look at some of those enforcement terms.
The first are the on-site inspections.
Treaties have included everything from surprise inspections (usually with 1-5 days notice, depending on treaty). Provisions requiring that each signing country allow for / accommodate an inspector from other countries while the destruction of weapons / delivery vehicles is taking place. Inspections from supposed neutral third parties (again, surprise or scheduled).
These treaties often have specific terms involving how many pictures can or cannot be taken, video or audial recordings, equipment for measuring radioactivity, specific requirements on the inspectors. And a lot more.
Telemetry data Includes data on test flights, including payload, location, expected locations, dates, times etc).
Satalites imaging has been both protected, and specifically enabled. Countries are supposed to do everything from keep delivery vehicles in the open (at specific times), notify each other as to the locations of nuclear arsenals, and more.
The full text of START can be found here https://1997-2001.state.gov/www/global/arms/starthtm/start/start1.html
Also worth a read, this is from one of the negotiators (Luitenant General Edward L Rowny).
Full text of SORT here
INF txt here
https://2009-2017.state.gov/t/avc/trty/102360.htm
Take a look at "Gassert, Philip (2020). The INF Treaty of 1987: A Reappraisal. Vandenhoeck & Ruprecht"