If you dig a little you can get the articles explaining better, but basically: yes, you have to have the weights, and a secret key… Given those, they split the words that might be chosen at each point into two groups and only choose from one group.

Later, you can check for the watermark by splitting the same way and seeing how many times the words in the text match the group that fits with the secret key. Normal text would hit a 50% match, but unlatered AI generated text would hit 100% match, so… somewhere higher than 50 you start to suspect AI involvement, possibly edited, and by the time you get over 99% it’s virtually certain that AI had some involvement - particularly if the sampling of words runs into the thousands.