Here are the official rules about selecting the Grand Prix contestants.. See especially 2.1a, 2,21 and 2.2h.
https://www.isu.org/docman-document...44-gp-general-announcement-2019-20-final/file
2.1 Seeded Skaters/Couples
a) Skaters/Couples who have placed 1 – 6 in each of the 4 disciplines (Men, Ladies, Pair Skating, Ice Dance) at the ISU World Figure Skating Championships 2019 will be considered to be Seeded Skaters/Couples and will be assigned, by draw, to skate in two events. ...
So officially the skaters do not get any say in the matter. (In practice they express their prferences and hope it works out that way.)
The document does not make clear if the skater's names are drawn out of a hat for each event, or if the federations draw for choosing first, second, third, etc. I think it is the federations order that is chosen by random draw, not the names of the skaters. (I might be wrong about this, but I don't think so.)
I think the whole process goes like this. I will use the men's field at Skate America as a clean-cut example.
Step 1. The USFSA makes its host picks. They go with
Nathan Chen
Jason Brown
To be decided.(later filled by Alexei Krasnozhon)
Step 2. They have to choose one of Chen, Hanyu or Zhou, but they have already chosen Chen, so they do not add anyone else to the roster in this round.
Step 3.Now they have to choose from among Uno, Jin and Kolyada. They choose Jin -- either because they are more afraid of the other two, or because they think the audience will like Jin, or because other federations have chosen before USA and Jin is the only one left.
Etc.
In the meantime they have asked the skaters for their input and made deals among the federations about who they will and won't select in the various rounds.
Something like that?