The first thing you may want to try is to see what other named users have already added and allowed you access to (including those items that are public).
If you Open an item from ArcGIS Online in ArcGIS Desktop you will have a new ArcMap Document and Table of Contents with the same name, and this will look and act very similar to the same Web Map opened in a browser in ArcGIS Online, Office, Sharepoint, or in ArcGIS Explorer on an iPhone etc. That is as long as you meet any security constraints, for instance if a data source is on a different network or is secured by ArcGIS Server and you do not have credentials then it just will not show.
In this case the map was authored by a named user in a browser and the points added by dragging a csv file onto one of Esri’s base maps and saved. Desktop has copied the points from the web map into a local file geodatabase, and copied the basemap tiles from one of Esri’s servers to the local display cache.
By using Details we can see some more of the metadata stored in ArcGIS Online, this time it some Australian Bureau of Statistics data uploaded to Esri’s servers by Map Data Services.
Following is a case where we have used Add to add a layer via an item that is a URL to an ArcGIS Server service hosted by Geoscience Australia.
Then we can Add the layer to the map, the description is held by ArcGIS Online but the content this time is from an ArcGIS Server not hosted by Esri.
You can choose to see what groups you have access to, and to extend the search beyond your organisation. Detail will show items you have access to within the group.
And lastly there are some Desktop functions that use available ArcGIS Online content; ArcGIS Online basemaps, World Traffic service as a layer and single address search.
To summarise the relationship to date, we have a marriage between a GIS Professional and anyone else with sufficient interest to either create content, document existing content, or define a map, all via ArcGIS Online. Note that so far we are just freely taking others content.
However there is one advantage of staying single and that is that the single line address search is free, geosearch vs geocoding.