This information might be out of date, but I think that the procedure is supposed to go something like this. Every member nation of the ISU puts its name in the hat for the random draw, except for countries that opt out because they do not have any judges in a particular discipline on the ISU international judges list. For major events this is done months in advance.
Then each National Federation that has been selected gets to choose internally which individual judge to send. In the 2002 pairs scandal. once France had made their choice, Madam La Goigne started receiving a raft of personal favors (birthday presents,invitations to parties, etc.) from both Canadian and Russian federation officials, LOL.
I have often been suspicious of the "randomness" of the initial draw (although it is duly witnessed and certified with appropriate ceremony) because the major feds seem to be chosen with much greater frequency than others.
The idea of having a quota by geographical region has been floated for decades, but never received much traction. The last thing the ISU needs is to give official approval to the formation of an Asia block, a Europe west of 20 degrees latitude bloc, etc.
For ISU Championships, the procedure is broadly still the same.
Only ISU judges can officiate at ISU Championships.
Worlds/Junior Worlds: draw is among countries with an entry in that discipline in the previous edition of said Championship. 13 countries are drawn. If there were less than 13 countries present in previous edition entered in the draw, there is an additional draw with all other ISU members to fill the remianing spots.
Europeans: same but with only European Feds.
4 Continents: only 9 countries are drawn. First draw: Feds with an entry the year before in that discipline. If they are less than 9, additional draw with all other 4 Continents Feds. If it is still not enough, a further draw with Euro Feds willing to serve.
After the draws, drawn Feds nominate the individual judges.
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Olympic
13 countries are drawn for each discipline.
First draw: done among countries who qualified a spot at Worlds
If needed, additional draw with countries which qualified a spot at the September's Qualifying competition (if they are not yet represented in the panel)
If needed, additional draw among countries present at previous edition's Worlds but not qualified for Olympics.
After the draws, drawn Feds nominate the individual judges.
Only ISU judges who have officited at least twice at ISU Championships/Olympic Qualifying Competition can serve at Olympics.
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In all other type of events, "international" judges can also serve.
Senior Grand Prix: there isn't a rule. However, the usual practices is that hosting Feds ask Feds with an entry to nominate a judge. Then, if needed, they fill the rest of the panel with whoever they want.
Junior Grand Prix: Feds with an entry are also entitled to nominate a judge.
Challenger Series and B events: same. Feds with an entry are asked to nominate a judge. If there are not enough nominated judges (in B events can happen), organizing committee invite the remaining judges.
Technical panels and Referees:
ISU Championships and Olympics: ISU pick up the officials. Hosting Fed can propose 2 technical specialists, controllers or referee for Singles and Pairs. And 1 for Ice Dance
Senior and Junior GPs: ISU pick up the officials.
Challenger Series: hosting Fed propose the composition of the tech panels and referees. ISU must confirm these names.
Other international events: hosts select the referees and tech panel members.