This can be achieved by first logging into the ArcGIS.com. Then you will need to go to an item which is a "service" (not just a shapefile or a webmap) in the My Content (such as a Feature Service which will be of "Features" type in the Type column).
Click the Feature Service link under Source:
You will be navigated to a web page which represents the Services Directory of the services you have published within your organization (which you have permissions to access to).
Look into the url:
Now you can just navigate to the
http://services.arcgis.com/%somecharactersnumbershere%/arcgis/rest/services/. From here you can work with your services much like you can from ArcGIS Server (including accessing the services in JSON).
If you need any additional info, refer to the Esri REST API web help pages here:
Working with users, groups, and items and Administering ArcGIS Online hosted services