As far as I know, there shouldn't be a difference between how action filters operate on Tableau Public and Tableau Desktop Professional. Some tests:
1. Download the workbook from Tableau Public and open in Tableau Desktop. Do the actions work correctly?
2. Publish from Tableau Desktop to Tableau Public, do the actions work correctly?
3. Is the behavior the same in all browsers?
Thanks. It turns out it was something else. I got it resolved. Thanks for the suggestions.