Twitter's terms of service don't allow tweet datasets to be published on the web, but they do allow tweet identifier datasets to be shared. This speaks to users rights as content creators, while also allowing researchers to share their data with others.
This site is a catalog of datasets that are publicly available on the web. If you would like to turn these tweet identifier datasets back into the original JSON first download the dataset and then use the Hydrator desktop application, or Twarc if you are comfortable working at the command line.
You can add your own datasets to the catalog by following these instructions.