Are you using hosted feature services in your collector app? If so, in ArcMap you can connect to your ArcGIS.com account by going to File > Sign In, and then you will be able to access your hosted feature services in the Catalog window under 'My Hosted Services'. You can then execute geoprocessing tool on these services. For example, you could use the Append tool to copy data from the hosted service to a local feature class.
Actually I am using ArcGIS Server feature services in my AGOL Map, and I download the data for offline use in the Collector app.
What I want to do is importing the data from the collector app before syncing them with the server