Is partial synchronisation possible between two databases on different servers?
The only built-in data synchronization technology that meets your requirements, I think, is Transactional Replication.
See the FAQ on replication:

How do I add or change indexes on publication and subscription databases?
Indexes can be added at the Publisher or Subscribers with no special considerations for replication (be aware that indexes can affect performance). CREATE INDEX and ALTER INDEX are not replicated, so if you add or change an index at, for example, the Publisher, you must make the same addition or change at the Subscriber if you want it reflected there.

This means you can have different objects (indexes, views, etc) on each replica.