No, you could share web maps all along. However, there is a difference between a web map and a map service. A web map contains a set of layers derrived from shapefiles, CSVs, KML, WMS, GPX, ArcGIS Server map services, and now, ArcGIS Online hosted map services. There are some limitations in terms of the size of shapefiles, CSVs, KML, and GPX data that you can display in your web map.
If you want to create your own map services, you can now purchase a subscription to ArcGIS Online. This allows you to create web services out of your data. For example, you can publish your map from ArcGIS Desktop and turn it into a service. Previously, you had to have your own server machines and ArcGIS Server to do so.
Hope this helps,
Mike