Absolutely, it's possible. Preferable, even, when you are running and administering ArcGIS Server instances in the cloud and need to allow for users to edit data via Desktop; especially, where a lost connection could disrupt an editing session. We've been using Amazon Web Services to host almost all of our server infrastructure for nearly 2 years. We are running ArcGIS Server Enterprise on Linux, backed with a PostgreSQL database server, also Linux. All our data in SDE is versioned and archive-enabled (for SDE and runtime mobile replication). Our cartographers do all their editing via Desktop, installed on Windows Server 2016 instances in the same AWS cloud environment. We also maintain all the MXDs and services on these boxes. Everyone accesses these Desktop boxes via RDP. Even better, if done correctly, it's very secure. I won't get into the details on security here, but I can say that our environment has passed multiple rigorous design reviews and external audits.
