From memory:
The tiles would've all had pre-determined letters. This makes the game on budget with the technology of the 80s. I'm not saying bar coding/RFID type stuff was impossible back then, but let's be realistic about what they would've likely used.
If there were two of the same letter in the word, a contestant picks two S, the word is assinine, the first time S is chosen it would fall into the 2nd space in the word, and the second time the S is chosen, it would fall into the 3rd space in the word.
This also follows if the player picked two A tiles, the first would fall into the 1st space in the word, the second would be the stopper.
As for the Scrabble Sprint, since two players had the same words to play (in the better versions of the show), the first two letters to appear were (I think my memory is correct here) always the same, and then everything else was random.
I never really knew how those letters were displayed as they had to be constantly shuffled around.
Scrabble was a show that almost always had the disclaimer read by Charlie Tuna during the credits, something like, "due to technical difficulties, portions of filming not affecting the outcome of the game were edited." I always figured out the equipment for the sprint rounds was tough to manage.