Yes, you can either export segment and trait level data or Customer Data Feeds (CDF) which is the raw data collected by the Data Collection Servers. Either your consultant or Client Care will be able to enable this for you. You will need to provide a sftp or s3 bucket that they deliver this data to.
I believe you are referring to subscribing to other traits and segments from Marketplace and shared audience. At present there is no log single report for all the subscribed data, and/or imported segments from shared audience.