Can I run my own Counterparty server?

Yes, although that requires a full bitcoin node and some technical knowledge.

Counterparty.io Docs section explaining how to run a Counterparty Node

Please refer to the Counterparty Docs page for details on how to setup a Counterparty Federated Node from the Official Counterparty Website.

If you are starting from scratch, it may take several days to download and index the bitcoin blockchain.