Generation 1 trading

After you got me interested, I did some research.
Apparently, it was distributed through a machine called the Mew Machine (Celebi in Generation II):

A person has posted on a forum about this machine, and his speculations are as follows:

I got a feeling that there are 2 Super Famicom's inside aswell as 2 Super Game Boy 2's that are linked with eachother.. but I'm not quite sure about the rest. Besides the monitor...

I'm afraid I can't find any concrete/official sources on this, but it seems like you were right about how it was controlled!

Additionally, I've found some sources stating that the Mew was simply traded (as I originally seemed to remember) in some regions. This was done using a "special" gameboy and game version, loaded with several level 5 Mews. It appears to be a regular gameboy with Pokémon Blue, though somehow modified or loaded with the Mews.

In generation 1 and 2 the IVs went from 0 to 15 for each stat. And there were only 5 stats - Special was one stat, and would not be separated into Special Attack and Special Defence until Gen3.

However, the IV for HP was determined a little differently:

The HP IV is calculated by taking the least significant bit (the final binary digit) of the Attack, Defense, Speed, and Special IVs, then creating a binary string by placing them in that order. As such, a Pokémon with an odd-number Attack IV has 8 added to its HP IV, an odd-number Defense IV has 4 added, an odd-number Speed IV has 2 added, and an odd-number Special IV has 1 added.

Effectively, as long as Att/Def/Speed/Special were all odd numbered, the HP IV was 15 (max). So only four IVs stats were randomly determined.

The chance of getting a single max-IV stat is 1 in 16. The chance for two perfects IVs is 1 in 256. Three is 1 in 4,096. All four is 1 in 65,536. And if all four stats are at 15, the HP is automatically 15.

So the answer is 1 in 65,536 or if you prefer percentages; roughly 0.001526% of Pokemon caught in generation 1 will have perfect IVs

Also keep in mind that in addition to those odds, the earlier generations did not have an NPC to check if IVs were maxed, so unless you're running the game on an emulator that will let you see the stats, you'll need to calculate them with the help of a IV/DV calculator.

As pointed out in the comments, the above is true ONLY if we assume that the random numbers are truly random. However, for a number of reasons, random numbers generated by computer programs are often not 100% random. This remains true for Pokemon RBY.

As it turns out; the way in which the Gen 1 games determine stats for random encounters means that not all 65,536 stat combinations are actually possible, and the number of available combinations depends on the encounter rate of the area (how many steps before an encounter occurs). At best there are about 7168 stat combinations possible (it varies depending on encounter rate), so the best possible stats have approximately a 1 in 7168 (0.014%) chance of being encountered.

And the icing on the cake is; a perfect 15/15/15/15 is not even on the list for possible stat spreads of a pokemon with an encounter rate of 25. In most cases the best you can hope for, it seems, is 15/15/14/15 or 15/15/15/7.

Only non-wild encounters can have max stats