It's not that the actual ML computation is done client side -- that would be way too slow. With Watson and many other ML platforms, you're leveraging someone else's (IBM, Google, Microsoft, Amazon, etc.) computing power.

Some people have made javascript-based ML models to run in the browser (I think some were made for this course[1]), but these are for educational purposes rather than actual use.

[1]: https://cs231n.github.io/