On our end I use SSMS directly from my system to reach all of our systems, and on the few off-site systems I can VPN into those and still use SSMS. Red Gate SQL Prompt has spoiled me, and given I have it locally it's nice to have regardless of which server I'm connected to.
As for authentication, my domain account is not a sysadmin on all of our servers, specially on vendor maintained servers, so on some i have to log in but most I can just use trusted authentication.
I honestly don't care to use SQL over RDP mainly because it's slow and we don't have SQL Prompt installed on any of the servers. Yeah I can get by without it, did for MANY years, but it's still nice to have