The data is not available for download directly from the map. You can, however, add CSV and Shapefile items and make these public. These items would be available for download. You could somehow provide links to them.
Be aware that a web map can support roughly 1000 features uploaded into it from Shapefiles and CSVs. That's 1000 features total, not per CSV or Shapefile. If you need to upload more data, you would need to get a subscription and create services from your data.
Thanks,
Mike