Can anyone (my admin, service provider, Virgil Security, etc.) decrypt the data I'm sending or receiving?

No. Encrypted data can only be decrypted with the private key corresponding to the public key it was encrypted with. 

In order to let several users decrypt the same encrypted data, specify all participants' public keys during encryption. 

Take a look at examples how to encrypt data for multiple participants, choose your case and try:

• My users have Virgil Cards at Virgil Cards Service.
• My users don't have Virgil Cards. I only use Virgil Crypto Library.